evaluating the lifetime performance index based on the bayesian

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 547209 13 pageshttpdxdoiorg1011552013547209

Research ArticleEvaluating the Lifetime Performance Index Based onthe Bayesian Estimation for the Rayleigh Lifetime Products withthe Upper Record Values

Wen-Chuan Lee1 Jong-Wuu Wu2 Ching-Wen Hong3 and Shie-Fan Hong2

1 Department of International Business Chang Jung Christian University Tainan 71101 Taiwan2Department of Applied Mathematics National Chiayi University 300 Syuefu Road Chiayi City 60004 Taiwan3Department of Information Management Shih Chien University Kaohsiung Campus Kaohsiung 84550 Taiwan

Correspondence should be addressed to Jong-WuuWu jwwumailncyuedutw

Received 20 October 2012 Accepted 17 December 2012

Academic Editor Chong Lin

Copyright copy 2013 Wen-Chuan Lee et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Quality management is very important for many manufacturing industries Process capability analysis has been widely applied inthe field of quality control to monitor the performance of industrial processes Hence the lifetime performance index119862

119871is utilized

to measure the performance of product where 119871 is the lower specification limit This study constructs a Bayesian estimator of 119862119871

under a Rayleigh distribution with the upper record values The Bayesian estimations are based on squared-error loss functionlinear exponential loss function and general entropy loss function respectively Further the Bayesian estimators of 119862

119871are utilized

to construct the testing procedure for 119862119871based on a credible interval in the condition of known 119871 The proposed testing procedure

not only can handle nonnormal lifetime data but also can handle the upper record values Moreover the managers can employthe testing procedure to determine whether the lifetime performance of the Rayleigh products adheres to the required level Thehypothesis testing procedure is a quality performance assessment system in enterprise resource planning (ERP)

1 Introduction

Process capability analysis is an effective means to measurethe performance and potential capabilities of a process Pro-cess capability indices (PCIs) are utilized to assess whetherproduct quality meets the required level in manufacturingindustries For instance the lifetime of electronic compo-nents exhibits a larger-the-better type of quality character-istic Montgomery [1] proposed the process capability index119862119871to evaluate the lifetime performance of electronic com-

ponents where 119871 is the lower specification limit Tong et al[2] constructed the uniformly minimum variance unbiasedestimator (UMVUE) of119862

119871and proposed a hypothesis testing

procedure for the complete sample from a one-parameterexponential distribution In addition the lifetime perfor-mance index 119862

119871also is applied to evaluate the lifetime of

product in the censored sample For instance Hong et al [34] constructed the lifetime performance index 119862

119871to evaluate

business performance under the right type II censored sample

and proposed a confidence interval for Paretorsquos distributionWu et al [5] proposed a hypothesis testing procedure basedon a maximum likelihood estimator (MLE) of 119862

119871to evaluate

the product quality for two-parameter exponential distri-bution under the right type II censored sample Lee et al[6] also proposed a hypothesis testing procedure based ona MLE of 119862

119871to evaluate product quality for exponential

distribution under the progressively type II right censoredsample Lee et al [7] also constructed an UMVUE of 119862

119871

and developed a testing procedure for the performance indexof products with the exponential distribution based on thetype II right censored sample All of the above 119862

119871have been

utilized to evaluate the quality performance for complete dataand censored data Nevertheless record values often arise inindustrial stress testing and other similar situations

Record values and the associated statistics are of interestand important in many real-life applications In industry andreliability studies many products fail under stress For exam-ple a battery dies under the stress of time But the precise

2 Journal of Applied Mathematics

breaking stress or failure point varies even among identicalitems Hence in such experiments measurements may bemade sequentially and only the record values (lower or upper)are observed Record values arise naturally in many real-life applications involving data relating to weather sportseconomics and life tests According to themodel of Chandler[8] there are some situations in lifetime testing experimentswhere the failure time of a product is recorded if it exceedsall preceding failure times These recorded failure times arethe upper record value sequence In general let 119883

1 1198832

be a sequence of independent and identically distributed(iid) random variables having the same distribution asthe (population) random variable 119883 with the cumulativedistribution function (cdf) 119865(119909) and the probability densityfunction (pdf) 119891(119909) An observation 119883

119895will be called an

upper record value if it exceeds in value all of the precedingobservations that is if119883

119895gt 119883119894 for every 119894 lt 119895 The sequence

of record times 119879119899 119899 ge 1 is defined as follows

Let 1198791= 1 with probability 1 for 119899 ge 2 119879

119899= min119895

119883119895gt 119883119879119899minus1

A sequence of upper record values 119883119880(119899) is

then defined by

119883119880(119899)

= 119883119879119899

119899 = 1 2 (1)

Let 119883sim= 119883119880(1) 119883119880(2) 119883

119880(119899) be the first 119899 upper record

values arising from a sequence of iid random variables1198831 1198832 with cdf 119865(119909)

In this study we consider the case of the upper recordvalues instead of complete data or censored data In orderto evaluate the lifetime performance of nonnormal datawith upper record values the lifetime performance index 119862

119871

is utilized to measure product quality under the Rayleighdistribution with the upper record values The Rayleighdistribution is a nonnormal distribution and a special case ofthe Weibull distribution which provides a population modeluseful in several areas of statistics including life testing andreliability whose age with time as its failure rate is a linearfunction of time Bhattacharya and Tyagi [9] mentioned thatin some clinical studies dealing with cancer patients thesurvival pattern follows the Rayleigh distribution Cliff andOrd [10] showed that the Rayleigh distribution arises as thedistribution of the distance between an individual and itsnearest neighbor when the special pattern is generated by thePoisson process Dyer and Whisenand [11] demonstrated theimportance of this distribution in communication engineer-ing (also see Soliman and Al-Aboud [12])The pdf and cdfof the Rayleigh distribution are given respectively by

119891 (119909) =119909

1205792119890minus1199092

21205792

(2)

119865 (119909) = 1 minus 119890minus1199092

21205792

(3)

where 119909 gt 0 and 120579 gt 0 respectively When119883 comes from theRayleigh distribution the mean of 119883 is 119864(119883) = radic1205872120579 andthe standard deviation of119883 is 120590 = radic(4 minus 120587)2120579

Bayesian and non-Bayesian approaches have been used toobtain the estimators of the parameter 120579 under the Rayleigh

distribution with the upper record values Soliman and Al-Aboud [12] compared the performance of the Bayesian esti-mators with non-Bayesian estimators such as the MLE andthe best linear unbiased (BLUE) estimatorThe Bayesian esti-mators are developed under symmetric and nonsymmetricloss functions The symmetric loss function is squared-error(SE) loss function The nonsymmetric loss function includeslinear exponential (LINEX) and general entropy (GE) lossfunctions In recent decades the Bayesian viewpoint hasreceived frequent attention for analyzing failure data andother time-to-event data and has been often proposed as avalid alternative to traditional statistical perspectives TheBayesian approach to estimation of the parameters andreliability analysis allows prior subjective knowledge onlifetime parameters and technical information on the failuremechanism as well as experimental data Bayesian methodsusually require less sample data to achieve the same qualityof inferences than methods based on sampling theory (alsosee [12])

The main aim of this study will construct the Bayesianestimator of 119862

119871under a Rayleigh distribution with upper

record values The Bayesian estimators of 119862119871are developed

under symmetric and nonsymmetric loss functions Theestimators of 119862

119871are then utilized to develop a credible

interval respectively The new testing procedures of credibleinterval for 119862

119871can be employed by managers to assess

whether the products performance adheres to the requiredlevel in the condition of known 119871 The new proposed testingprocedures can handle nonnormal lifetime data with upperrecord values Moreover we will evaluate the performance ofthe new proposed testing procedures with Bayesian and non-Bayesian approaches

The rest of this study is organized as follows Section 2introduces some properties of the lifetime performanceindex for lifetime of product with the Rayleigh distributionSection 3 discusses the relationship between the lifetimeperformance index and conforming rate Section 4 developsa hypothesis testing procedure for the lifetime performanceindex119862

119871with the non-Bayesian approach Section 5 develops

a hypothesis testing procedure for the lifetime performanceindex 119862

119871with the Bayesian approach Section 6 discusses the

MonteCarlo simulation algorithmof confidence (or credible)level One numerical example and concluding remarks aremade in Sections 7 and 8 respectively

2 The Lifetime Performance Index

Montgomery [1] has developed a process capability index 119862119871

to measure the larger-the-better quality characteristic Then119862119871is defined by

119862119871=120583 minus 119871

120590 (4)

where 120583 120590 and 119871 are the process mean the process standarddeviation and the lower specification limit respectively

To assess the product performance of products 119862119871can

be defined as the lifetime performance index If 119883 comesfrom theRayleigh distribution then the lifetime performance

Journal of Applied Mathematics 3

index 119862119871can be rewritten as

119862119871=120583 minus 119871

120590=radic1205872120579 minus 119871

radic(4 minus 120587) 2120579= radic

120587

4 minus 120587minus radic

2

4 minus 120587

119871

120579

minusinfin lt 119862119871lt radic

120587

4 minus 120587

(5)

where 120583 = 119864(119883) = radic1205872120579 is the process mean 120590 =

radicVar(119883) = radic(4 minus 120587)2120579 is the process standard deviationand 119871 is the lower specification limit

3 The Conforming Rate

If the lifetime of a product 119883 exceeds the lower specificationlimit 119871 then the product is defined as a conforming productThe ratio of conforming products is known as the conformingrate and can be defined as

119875119903= 119875 (119883 ge 119871) = 119890

minus(12)(radic1205872minusradic(4minus120587)2119862119871)2

minus infin lt 119862119871lt radic120587 (4 minus 120587)

(6)

Obviously a strictly increasing relationship existsbetween the conforming rate 119875

119903and the lifetime

performance index 119862119871 Table 1 lists various 119862

119871values and

the corresponding conforming rate 119875119903by using STATISTICA

software [13] (also see [14])For the 119862

119871values which are not listed in Table 1 the

conforming rate 119875119903can be obtained through interpolation In

addition since a one-to-one mathematical relationship existsbetween the conforming rate119875

119903and the lifetime performance

index 119862119871 Therefore utilizing the one-to-one relationship

between 119875119903and 119862

119871 lifetime performance index can be

a flexible and effective tool not only evaluating productperformance but also estimating the conforming rate 119875

119903

4 Testing Procedure for the LifetimePerformance Index 119862

119871with

Non-Bayesian Approach

This section will apply non-Bayesian approach to construct amaximum likelihood estimator (MLE) of119862

119871under aRayleigh

distribution with upper record values The MLE of 119862119871is

then utilized to develop a new hypothesis testing procedurein the condition of known 119871 Assuming that the requiredindex value of lifetime performance is larger than 119888

0 where

1198880denotes the target value the null hypothesis 119867

0 119862119871le 1198880

and the alternative hypothesis1198671 119862119871gt 1198880are constructed

Based on the new hypothesis testing procedure the lifetimeperformance of products is easy to assess

Let 119883 be the lifetime of such a product and 119883 hasa Rayleigh distribution with the pdf as given by (2)Let 119883sim

= (119883119880(1) 119883119880(2) 119883

119880(119899)) be the first 119899 upper

record values arising from a sequence of iid Rayleigh

variables with pdf as given by (2) Since the joint pdf of(119883119880(1) 119883119880(2) 119883

119880(119899)) is

119891 (119909119880(119899))

119899minus1

prod

119894=1

119891 (119909119880(119894))

1 minus 119865 (119909119880(119894)) (7)

where 119891(119909) and 119865(119909) are the pdf and cdf of 119883 respec-tively (also see [12 15 16]) So the likelihood function of(119883119880(1) 119883119880(2) 119883

119880(119899)) is given as

119871 (120579 | 119909119880(1) 119909

119880(119899)) =

119899

prod

119894=1

119909119880(119894)

1205792 exp(minus

1199092

119880(119899)

21205792) (8)

The natural logarithm of the likelihood function with (8) is

ln 119871 (120579 | 119909119880(1) 119909

119880(119899)) =

119899

sum

119894=1

ln (119909119880(119894)) minus 2119899 ln 120579 minus

1199092

119880(119899)

21205792

(9)

Upon differentiating (9) with respect to 120579 and equating theresult to zero theMLE of the parameter 120579 can be shown to be

120579MLE =119883119880(119899)

radic2119899(10)

(also see [12]) By using the invariance of MLE (see Zehna[17]) the MLE of 119862

119871can be written as given by

119862119871MLE = radic

120587


2

4 minus 120587

119871

120579MLE

= radic120587


2

4 minus 120587sdot 119871 sdot

radic2119899

119883119880(119899)

(11)

Construct a statistical testing procedure to assess whetherthe lifetime performance index adheres to the required levelThe one-sided confidence interval for 119862

119871is obtained using

the pivotal quantity 1198832119880(119899)1205792 By using the pivotal quantity

1198832

119880(119899)1205792 given the specified significance level 120572 the level

100(1 minus 120572) one-sided confidence interval for 119862119871can be

derived as followsSince the pivotal quantity 1198832

119880(119899)1205792sim 1205942

2119899 and 1205942

21198991minus120572

function which represents the lower 1 minus 120572 percentile of 12059422119899

then

119875(1198832

119880(119899)

1205792le 1205942

21198991minus120572) = 1 minus 120572

997904rArr 119875(119862119871ge radic

120587

4 minus 120587minus(radic

120587

4 minus 120587minus119862119871MLE)(

1205942

21198991minus120572

2119899)

12

)

= 1 minus 120572

(12)

where 119862119871= radic120587(4 minus 120587) minus radic2(4 minus 120587)(119871120579) From (12) we


Table 1 The lifetime performance index 119862119871versus the conforming rate 119875

119903

119862119871

119875119903

119862119871

119875119903

119862119871

119875119903

minusinfin 0000000 040 0611832 120 0896627minus450 0000147 045 0631685 125 0909965minus400 0000551 050 0651484 130 0922511minus350 0001858 055 0671182 135 0934227minus300 0005628 060 0690734 140 0945076minus250 0015308 065 0710094 145 0955027minus200 0037404 070 0729213 150 0964047minus150 0082094 075 0748044 155 0972109minus100 0161849 080 0766539 160 0979187minus050 0286621 085 0784648 165 0985259000 0455938 090 0802324 170 0990306015 0513214 095 0819518 175 0994310020 0532718 100 0836183 180 0997261025 0552370 105 0852271 185 0999147030 0572132 110 0867738 190 0999963035 0591967 115 0882538 191 0999997

obtain that a 100(1 minus 120572) one-sided confidence interval for119862119871is

119862119871ge radic

120587

4 minus 120587minus (radic

120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(13)

where119862119871MLE is given by (11)Therefore the 100(1minus120572) lower

confidence interval bound for 119862119871can be written as

LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(14)

where 119862119871MLE 120572 and 119899 denote the MLE of 119862

119871 the specified

significance level and the upper record sample of sizerespectively

The managers can use the one-sided confidence intervalto determine whether the product performance adheres tothe required levelThe proposed testing procedure of119862

119871with

119862119871MLE can be organized as follows

Step 1 Determine the lower lifetime limit 119871 for productsand the performance index value 119888

0 then the testing null

hypothesis 1198670 119862119871le 1198880and the alternative hypothesis

1198671 119862119871gt 1198880are constructed

Step 2 Specify a significance level 120572

Step 3 Given the upper record values 119909sim= (119909119880(1) 119909119880(2)

119909119880(119899)) the lower lifetime limit 119871 and the significance level 120572

then we can calculate the 100(1 minus 120572) one-sided confidenceinterval [LBMLEinfin) for 119862119871 where LBMLE as the definition of(14)

Step 4 The decision rule of statistical test is provided asfollows

(1) if the performance index value 1198880notin [LBMLEinfin) then

we will reject 1198670 It is concluded that the lifetime

performance index of products meets the requiredlevel

(2) if the performance index value 1198880isin [LBMLEinfin) then

we do not reject 1198670 It is concluded that the lifetime

performance index of products does not meet therequired level

5 Testing Procedure forthe Lifetime Performance Index 119862

119871with

the Bayesian Approach

This section will apply the Bayesian approach to constructan estimator of 119862


record values The Bayesian estimators are developed undersymmetric and nonsymmetric loss functionsThe symmetricloss function is squared-error (SE) loss function The non-symmetric loss function includes linear exponential (LINEX)and general entropy (GE) loss functions The Bayesian esti-mator of 119862

119871is then utilized to develop a new hypothesis

testing procedure in the condition of known 119871 Assumingthat the required index value of lifetime performance is largerthan 119888

0 where 119888

0denotes the target value the null hypothesis

1198670 119862119871le 1198880and the alternative hypothesis119867

1 119862119871gt 1198880are

constructed Based on the new hypothesis testing procedurethe lifetime performance of products is easy to assess

51 Testing Procedure for the Lifetime Performance Index119862119871with the Bayesian Estimator under Squared-Error Loss

Function Let 119883 be the lifetime of such a product and 119883has a Rayleigh distribution with the pdf as given by (2)We consider the conjugate prior distribution of the formwhich is defined by the square-root inverted-gamma density


as follows

1205871 (120579) =

119886119887

Γ (119887) 2119887minus1120579minus2119887minus1

119890minus1198862120579

2

(15)

where 120579 gt 0 119886 gt 0 and 119887 gt 0

119864 (120579) = (119886

2)

12Γ (119887 minus 12)

Γ (119887)

Var (120579) = 119886

2 (119887 minus 1)minus (

119886

2) [Γ (119887 minus 12)

Γ (119887)]

2

119887 gt 1

(16)

With record values 119899 products (or items) take place ontest Consider (119883

119880(1) 119883119880(2) 119883

119880(119899)) is the upper record

values We can obtain that the posterior pdf of 120579 | (119883119880(1)

119883119880(2) 119883

119880(119899)) is given as

1199011(120579 | 119909

119880(1) 119909

119880(119899))

=119871 (120579 | 119909

119880(1) 119909

119880(119899)) 1205871 (120579)

intinfin

0119871 (120579 | 119909

119880(1) 119909

119880(119899)) 1205871 (120579) 119889120579

=

120579minus2(119899+119887)minus1

(1199092

119880(119899)+ 119886)(119899+119887)

119890minus(119909119880(119899)+119886)2120579

2

Γ (119899 + 119887) 2119899+119887minus1

(17)

where 120579 gt 0 and the likelihood function 119871(120579 | 119909119880(1)

119909119880(119899)) as (8)Let 119905 = 1199092

119880(119899)+119886 then the posterior pdf can be rewritten

as

1199011(120579 | 119909

119880(1) 119909

119880(119899)) =

120579minus2(119899+119887)minus1

119905(119899+119887)

119890minus1199052120579

2

Γ (119899 + 119887) 2119899+119887minus1 (18)

and we can show that 1199051205792|119909sim

=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

The Bayesian estimator 120579BS of 120579 based on the squarederror (SE) loss function 119871(120579lowast 120579) = (120579lowast minus120579)2 can be derived asfollows

119864 (120579 | 119909119880(1) 119909

119880(119899))

= int

infin

0

1205791199011(120579 | 119909

119880(1) 119909

119880(119899)) 119889120579

=Γ (119899 + 119887 minus 12)

Γ (119899 + 119887)(119905

2)

12

(19)

where 119905 = 1199092119880(119899)

+ 119886 119899 + 119887 minus 12 gt 0Hence the Bayesian estimator of 120579 based on SE loss

function is given by

120579BS =Γ (119899 + 119887 minus 12)

Γ (119899 + 119887)(119879

2)

12

(20)

where 119879 = 1198832

119880(119899)+ 119886 119899 + 119887 minus 12 gt 0 119886 and 119887 are the

parameters of prior distribution with density as (15)The lifetime performance index 119862

119871of Rayleigh products

can be written as

119862119871= radic

120587


2

4 minus 120587

119871

120579 minusinfin lt 119862

119871lt radic

120587

4 minus 120587 (21)

By using (5) and the Bayesian estimator 120579BS as (20) 119862119871BS

based on the Bayesian estimator 120579BS of 120579 is given by

119862119871BS = radic

120587


2

4 minus 120587

119871

120579BS

= radic120587


2

4 minus 120587sdot 119871 sdot [

Γ (119899 + 119887 minus 12)

Γ (119899 + 119887)(119879

2)

12

]

minus1

(22)

where 119879 = 1198832


parameters of prior distribution with density as in (15)We construct a statistical testing procedure to assess

whether the lifetime performance index adheres to therequired level The one-sided credible confidence interval for119862119871are obtained using the pivotal quantity 1199051205792|

119909sim

=(119909119880(1)119909119880(119899))

By using the pivotal quantity 1199051205792|119909sim

=(119909119880(1)119909119880(119899)) given the

specified significance level 120572 the level 100(1minus120572) one-sidedcredible interval for 119862

119871can be derived as follows

Since the pivotal quantity 1199051205792|119909sim

=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572

function which represents the lower 1 minus 120572percentile of 1205942

2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

| 119909sim)

= 1 minus 120572

(23)

where 119862119871as the definition of (5)

From (23) we obtain that a 100(1minus120572)one-sided credibleinterval for 119862

119871is given by

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

(24)

where 119862119871BS is given by (22)

Therefore the 100(1 minus 120572) lower credible interval boundfor 119862119871can be written as

LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

(25)


where 119862119871BS is given by (22) and 119887 is a parameter of prior

distribution with density as (15)The managers can use the one-sided credible interval to

determine whether the product performance adheres to therequired level The proposed testing procedure of 119862

119871with

119862119871BS can be organized as follows






Step 3 Given the parameters 119886 and 119887 of prior distributionthe upper record values 119909

sim= (119909

119880(1) 119909119880(2) 119909

119880(119899)) the

lower lifetime limit 119871 and the significance level 120572 then wecan calculate the 100(1 minus 120572) one-sided credible interval[LBBSinfin) for 119862119871 where LBBS as the definition of (25)


(1) if the performance index value 1198880notin [LBBSinfin) then



(2) if the performance index value 1198880isin [LBBSinfin) then



52 Testing Procedure for the Lifetime Performance Index 119862119871

with the Bayesian Estimator under Linear Exponential LossFunction Let 119883 be the lifetime of such a product and 119883has a Rayleigh distribution with the pdf as given by (2) Weconsider the linear exponential (LINEX) loss function as (alsosee [18ndash23])

119871 (Δ) prop 119890119888Δminus 119888Δ minus 1 (26)

where Δ = (120579lowast minus 120579) 119888 = 0In this paper we suppose that

Δ1= (

120579lowast

120579)

2

minus 1 (27)

where 120579lowast is an estimator of 120579By using (26)-(27) the posterior expectation of LINEX

loss function 119871(Δ1) is

119864 [119871 (Δ1) | 119909sim] = 119890minus119888119864 [119890119888(120579lowast

120579)2 | 119909sim]

minus 119888119864 [(120579lowast

120579)

2

minus 1 | 119909sim] minus 1

(28)

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

The value of 120579lowast that minimizes 119864[119871(Δ1) | 119909sim] denoted by

120579BL is obtained by solving the equation

119889119864 [119871 (Δ1) | 119909sim]

119889120579lowast= 119890minus119888119864[119890119888(120579lowast

120579)2 sdot2119888120579lowast

1205792| 119909sim]

minus 119888119864(2120579lowast

1205792| 119909sim) = 0

(29)

that is 120579BL is the solution to the following equation

119864[120579BL1205792119890119888(120579BL120579)2 | 119909

sim] = 119890119888119864(

120579BL1205792

| 119909sim) (30)

By using (18) and (30) we have

2120579BL (119899 + 119887) sdot119905119899+119887

(119905 minus 21198881205792

BL)119899+119887+1

= 119890119888120579BL

2 (119899 + 119887)

119905

where 119905 = 1199092119880(119899)

+ 119886

(31)

Hence the Bayesian estimator of 120579 under the LINEX lossfunction is given by

120579BL = [119879

2119888(1 minus 119890

minus119888(119899+119887+1))]

12

(32)

where 119879 = 1198832

119880(119899)+ 119886 119886 and 119887 are the parameters of prior

distribution with density as (15)By using (5) and the Bayesian estimator 120579BL as (32) 119862119871BL

based on the Bayesian estimator 120579BL of 120579 is given by

119862119871BL = radic

120587


2

4 minus 120587

119871

120579BL

= radic120587


2

4 minus 120587119871[119879

2119888(1 minus 119890

minus119888(119899+119887+1))]

minus12

(33)

where 119879 = 1198832


distribution with density as (15)We construct a statistical testing procedure to assess

whether the lifetime performance index adheres to therequired level The one-sided credible confidence interval for119862119871is obtained using the pivotal quantity 1199051205792|

119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the





=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

| 119909sim)

= 1 minus 120572

(34)



119871is

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

(35)

where 119862119871BL is given by (33)


LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

(36)

where 119862119871BL is given by (33) and 119887 is a parameter of prior


determine whether the product performance attains to therequired level The proposed testing procedure of 119862

119871with

119862119871BL can be organized as follows






Step 3 Given the parameters 119886 and 119887 of prior distribution theparameter 119888 of LINEX loss function the upper record values119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899)) the lower lifetime limit 119871 and the

significance level 120572 then we can calculate the 100(1 minus 120572)one-sided credible interval [LBBLinfin) for 119862119871 where LBBL asthe definition of (36)


(1) if the performance index value 1198880notin [LBBLinfin) then



(2) if the performance index value 1198880isin [LBBLinfin) then



53 Testing Procedure for the Lifetime Performance Index119862119871with the Bayesian Estimator under General Entropy Loss

Function Let 119883 be the lifetime of such a product and 119883has a Rayleigh distribution with the pdf as given by (2)We consider the general entropy (GE) loss function (also see[12 20])

119871 (120579lowast 120579) prop (

120579lowast

120579)

119902

minus 119902 log(120579lowast

120579) minus 1 (37)

whose minimum occurs at 120579lowast = 120579The loss function is a generalization of entropy loss used

by several authors (eg Dyer and Liu [24] Soliman [25] andSoliman and Elkahlout [26]) where the shape parameter 119902 =1 When 119902 gt 0 a positive error (120579lowast gt 120579) causes more seriousconsequences than a negative error The Bayesian estimator120579BG of 120579 under GE loss function is given by

120579BG = [119864 (120579minus119902| 119909119880(1) 119909

119880(119899))]minus1119902

(38)

By using (18) and (38) then the Bayesian estimator 120579BG of120579 is derived as follows

119864 (120579minus119902| 119909119880(1) 119909

119880(119899))

= int

infin

0

120579minus1199021199011(120579 | 119909

119880(1) 119909

119880(119899)) 119889120579

= (119905

2)

minus1199022

sdotΓ (119899 + 119887 + 1199022)

Γ (119899 + 119887)

(39)

where 119905 = 1199092119880(119899)

+ 119886 Hence the Bayesian estimator 120579BG of 120579under the GE loss function is given by

120579BG = (119879

2)

12

[Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(40)

where 119879 = 1198832


distribution with density as (15)By using (5) and the Bayesian estimator 120579BG as (40)119862

119871BGbased on the Bayesian estimator 120579BG of 120579 is given by

119862119871BG = radic

120587


2

4 minus 120587

119871

120579BG

= radic120587


2

4 minus 120587sdot 119871

sdot [(119879

2)

minus1199022Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

1119902

(41)


where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the




=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times[Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

| 119909sim)

= 1 minus 120572

(42)


From (42) we obtain that a 100(1 minus 120572) one-sidedcredible interval for 119862

119871is

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(43)

where 119862119871BG is given by (41)


LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(44)

where 119862119871BG is given by (41) 120572 is the specified significance

level and 119887 is a parameter of prior distribution with densityas (15)

The managers can use the one-sided credible interval todetermine whether the product performance attains to therequired level The proposed testing procedure of 119862

119871with

119862119871BG can be organized as follows






Step 3 Given the parameters 119886 and 119887 of prior distributionthe parameter 119902 of GE loss function the upper record values119909sim= (119909119880(1) 119909119880(2) 119909


significance level 120572 then we can calculate the 100(1 minus 120572)one-sided credible interval [LBBGinfin) for 119862119871 where LBBG asthe definition of (44)


(1) if the performance index value 1198880notin [LBBGinfin) then



(2) if the performance index value 1198880isin [LBBGinfin) then



6 The Monte Carlo Simulation Algorithm ofConfidence (or Credible) Level

In this section we will report the results of a simulationstudy for confidence (or credible) level (1 minus 120572) based on a100(1 minus 120572) one-sided confidence (or credible) interval ofthe lifetime performance index 119862

119871 We considered 120572 = 005

and then generated samples from the Rayleigh distributionwith pdf as in (2) with respect to the record values

(i) The Monte Carlo simulation algorithm of confidencelevel (1 minus 120572) for 119862

119871under MLE is given in the

following steps

Step 1 Given 119899 119886 119887 119871 and 120572 where 119886 gt 0 119887 gt 0

Step 2 For the values of prior parameters (119886 119887) use (15) togenerate 120579 from the square-root inverted-gamma distribu-tion

Step 3(a) The generation of data (119885

1 1198852 119885

119899) is by standard

exponential distribution with parameter 1(b) Set1198842

1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(45)

(1198841 1198842 119884

119899) are the upper record values from the

Rayleigh distribution(c) The value of LBMLE is calculated by

LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(46)


where 119862119871MLE is given by (11) and 1205942

21198991minus120572function

which represents the lower 1 minus 120572 percentile of 12059422119899

(d) If 119862119871ge LBMLE then count = 1 else count = 0

Step 4

(a) Step 3 is repeated 1000 times(b) The estimation of confidence level (1 minus 120572) is (1 minus 120572) =

(total count)1000

Step 5

(a) Repeat Step 2ndashStep 4 for 100 times then we can getthe 100 estimations of confidence level as follows

(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (47)

(b) The average empirical confidence level 1 minus 120572

of (1 minus 120572)119894 119894 = 1 100 that is 1 minus 120572 =

(1100)sum100

119894=1(1 minus 120572)

119894

(c) The sample mean square error (SMSE) of(1 minus 120572)

1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2

The results of simulation are summarized in Table 2 basedon 119871 = 10 the different value of sample size 119899 priorparameter (119886 119887) = (2 5) (6 15) and (2 2) at 120572 = 005respectively The scope of SMSE is between 000413 and000593



Step 3

(a) The generation of data (1198851 1198852 119885



1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(48)

(1198841 1198842 119884


Rayleigh distribution(c) The values of LBBS LBBL and LBBG are calculated by

(25) (36) and (44) respectively(d) If119862

119871ge LB then count = 1 else count = 0 where LB =

LBBS or LBBL or LBBG

Step 4

(a) Step 3 is repeated 1000 times(b) The estimation of credible level (1 minus 120572) is (1 minus 120572) =

(total count)1000

Step 5

(a) Repeat Step 2ndashStep 4 for 100 times then we can getthe 100 estimations of credible level as follows

(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (49)

(b) The average empirical credible level 1 minus 120572 of (1 minus 120572)119894

119894 = 1 100 that is 1 minus 120572 = (1100)sum100119894=1(1 minus 120572)

119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2

Based on 119871 = 10 the different value of sample size119899 prior parameter (119886 119887) = (2 5) (6 15) (2 2) and 120572 =

005 the results of simulation are summarized in Tables 3ndash5under SE loss function LINEX loss function with parameter119888 = minus05 05 15 and GE loss function with parameter119902 = 3 5 8 respectively The following points can be drawn

(a) All of the SMSEs are small enough and the scope ofSMSE is between 000346 and 000467

(b) Fix the size 119899 comparison of prior parameter (119886 119887) =(2 5) (6 15) and (2 2) as follows

(i) the values of SMSEwith prior parameter (119886 119887) =(6 15) and (2 5) are smaller than (119886 119887) = (2 2)respectively

(ii) a comparison of prior parameter (119886 119887) = (2 5)and (2 2) that is fix prior parameter 119886 if priorparameter 119887 increases then it can be seen thatthe SMSE will decrease

(c) In Table 4 fix the size 119899 and the prior parameter (119886 119887)comparison the parameter of LINEX loss function 119888 =minus05 05 15 as followsthe values of SMSE with prior parameter 119888 = 05 15andminus15 are the same as the values of SMSE inTable 3

(d) In Table 5 fix the prior parameter (119886 119887) comparisonof GE loss function parameter 119902 = 3 5 8 as followsthe values of SMSE with prior parameter 119902 = 3 5 and8 are the same as the values of SMSE in Table 3

Hence these results from simulation studies illustrate thatthe performance of our proposed method is acceptableMoreover we suggest that the prior parameter (119886 119887) = (6 15)and (2 5) is appropriate for the square-root inverted-gammadistribution

7 Numerical Example

In this section we propose the new hypothesis testingprocedures to a real-life data In the following examplewe discuss a real-life data for 25 ball bearings to illustratethe use of the new hypothesis testing procedures in thelifetime performance of ball bearings The proposed testingprocedures not only can handle nonnormal lifetime data butalso can handle the upper record values


Table 2 Average empirical confidence level (1 minus 120572) for 119862119871under MLE when 120572 = 005

119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

5 093653 (000462) 093482 (000467) 093577 (000507)10 093643 (000449) 093581 (000413) 093450 (000593)15 093523 (000489) 093557 (000503) 093463 (000516)119899 denotes the sample size the values in parentheses are sample mean square error of (1 minus 120572)

Table 3 Average empirical credible level (1 minus 120572) for 119862119871under SE loss function when 120572 = 005

119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2


Table 4 Average empirical credible level (1 minus 120572) for 119862119871under LINEX loss function when 120572 = 005

119899 119888 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

minus05 094357 (000386) 093760 (000386) 093925 (000395)5 05 094357 (000386) 093760 (000386) 093925 (000395)

15 094357 (000386) 093760 (000386) 093925 (000395)minus05 094244 (000359) 093704 (000359) 093664 (000462)

10 05 094244 (000359) 093704 (000359) 093664 (000462)15 094244 (000359) 093704 (000359) 093664 (000462)minus05 094096 (000346) 093619 (000426) 093635 (000467)

15 05 094096 (000346) 093619 (000426) 093635 (000467)15 094096 (000346) 093619 (000426) 093635 (000467)

119899 denotes the sample size the values in parentheses are sample mean square error of (1 minus 120572)

Table 5 Average empirical credible level (1 minus 120572) for 119862119871under GE loss function when 120572 = 005

119899 119902 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

3 094357 (000386) 093760 (000386) 093925 (000395)5 5 094357 (000386) 093760 (000386) 093925 (000395)

8 094357 (000386) 093760 (000386) 093925 (000395)3 094244 (000359) 093704 (000359) 093664 (000462)

10 5 094244 (000359) 093704 (000359) 093664 (000462)8 094244 (000359) 093704 (000359) 093664 (000462)3 094096 (000346) 093619 (000426) 093635 (000467)

15 5 094096 (000346) 093619 (000426) 093635 (000467)8 094096 (000346) 093619 (000426) 093635 (000467)


Example The data is the failure times of 25 ball bearings inendurance testThe 25 observations are the number ofmillionrevolutions before failure for each of the ball bearings Thedata come from Caroni [27] as follows

6780 6780 6780 6864 3300 6864 9864 12804 42122892 4560 5184 5556 17340 4848 1788 9312 54124152 5196 12792 8412 10512 10584 and 6888

Raqab and Madi [28] and Lee [29] indicated that a one-parameter Rayleigh distribution is acceptable for these dataIn addition we also have a way to test the hypothesis thatthe failure data come from the Rayleigh distribution Thetesting hypothesis119867

0 119883 comes from a Rayleigh distribution

versus1198671 119883 does not come from a Rayleigh distribution is

constructed under significance level 120572 = 005From a probability plot of Figure 1 by using the Minitab

Statistical Software we can conclude the operational lifetimesdata of 25 ball bearings from the Rayleigh distribution whichis the Weibull distribution with the shape 2 (also see Lee etal [14]) For informative prior we use the prior information119864(120579) = 120579MLE ≐ 54834 and Var(120579) = 02 giving the priorparameter values as (119886 = 6014 119887 = 1001)

Here if only the upper record values have been observedthese are 119909

119880(119894) 119894 = 1 5(= 119899) = 6780 6864 9864

12804 17340


10010

99

90807060504030

20

10

5

32

1

Xi

Probability plot of XiWeibull-95 CI

()

Shape 2

Scale 8002N 25

AD 0431P -value 057

Figure 1 Probability plot for failure times of 25 ball bearings data

(1) The proposed testing procedure of 119862119871with 119862

119871MLE isstated as follows

Step 1 The record values from the above data are (119909119880(119894) 119894 =

1 2 3 4 5) = (6780 6864 9864 12804 17340)

Step 2 The lower specification limit 119871 is assumed to be 2337The deal with the product managersrsquo concerns regardinglifetime performance and the conforming rate 119875

119903of products

is required to exceed 80 percent Referring to Table 1 119862119871is

required to exceed 090Thus the performance index value isset at 119888

0= 090 The testing hypothesis119867

0 119862119871le 090 versus

1198671 119862119871gt 090 is constructed

Step 3 Specify a significance level 120572 = 005

Step 4 We can calculate that the 95 lower confidenceinterval bound for 119862

119871 where

LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

= radic120587


120587

4 minus 120587minus 126251254)(

1205942

10095

10)

12

= 103285

(50)So the 95 one-sided confidence interval for 119862

119871is

[LBMLEinfin) = [103285infin)

Step 5 Because of the performance index value 1198880= 090 notin

[LBMLEinfin) we reject the null hypothesis1198670 119862119871 le 090

Thus we can conclude that the lifetime performanceindex of data meets the required level

(2) The proposed testing procedures of 119862119871with 119862

119871BS119862119871BL and 119862119871BG are stated as follows


1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus


Step 3 Specify a significance level 120572 = 005 the parameter ofLINEX loss function 119888 = 05 and the parameter of GE lossfunction 119902 = 2

Step 4 We can calculate the 95 lower credible intervalbound for 119862

119871 where

LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1246074685)

timesΓ (5 + 1001 minus 12)

Γ (5 + 1001)(1205942

2(5+1001)095

2)

12

= 096984

(51)


So the 95 one-sided credible interval for 119862119871

is[LBBSinfin) = [096984infin)

LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

= radic120587


120587

4 minus 120587minus 1129561059)

times [1

2 (05)(1 minus 119890

minus05(5+1001+1)) 1205942

2(5+1001)095]

12

= 096984

(52)


is[LBBLinfin) = [096984infin)

LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Τ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1200432998)

times [Γ (5 + 1001 + 22)

Γ (5 + 1001)]

minus12

(1205942

2(5+1001)095

2)

12

= 096984

(53)


is[LBBGinfin) = [096984infin)


[LBBSinfin) = [LBBLinfin) = [LBBGinfin) we reject the nullhypothesis119867

0 119862119871le 090

Thus we can conclude that the lifetime performance index ofdata meets the required level In addition by using the priorparameter values (119886 119887) = (6 15) based on the simulationstudies we also obtain that 119888

0= 090 notin [LBBSinfin) =

[LBBLinfin) = [LBBGinfin) = 094033 So we also rejectthe null hypothesis 119867

0 119862119871le 090 Hence the lifetime

performance index of data meets the required level

8 Conclusions

Montgomery [1] proposed a process capability index 119862119871for

larger-the-better quality characteristic The assumption ofmost process capability is normal distribution but it is ofteninvalid In this paper we consider that Rayleigh distributionis the special case of Weibull distribution

Moreover in lifetime data testing experiments the exper-imenter may not always be in a position to observe the life

times of all the products put to test In industry and reliabilitystudies many products fail under stress for example anelectronic component ceases to function in an environmentof too high temperature and a battery dies under the stressof time So in such experiments measurement may be madesequentially and only the record values are observed

In order to let the process capability indices can beeffectively used This study constructs MLE and Bayesianestimator of 119862

119871under assuming the conjugate prior distribu-

tion and SE loss function LINEX loss function and GE lossfunction based on the upper record values from the Rayleighdistribution The Bayesian estimator of 119862

119871is then utilized to

develop a credible interval in the condition of known 119871This study also provides a table of the lifetime perfor-

mance index with its corresponding conforming rate basedon the Rayleigh distributionThat is the conforming rate andthe corresponding 119862

119871can be obtained

Further the Bayesian estimator and posterior distributionof 119862119871are also utilized to construct the testing procedure of

119862119871which is based on a credible interval If you want to test

whether the products meet the required level you can utilizethe proposed testing procedure which is easily applied andan effectively method to test in the condition of known 119871Numerical example is illustrated to show that the proposedtesting procedure is effectively evaluating whether the trueperformance index meets requirements

In addition these results from simulation studies illus-trate that the performance of our proposedmethod is accept-able According to SMSE the Bayesian approach is smallerthan the non-Bayesian approach we suggest that the Bayesianapproach is better than the non-Bayesian approachThen theSMSEs of prior parameters (119886 119887) = (6 15) and (2 5) aresmaller than the SMSE of prior parameter (119886 119887) = (2 2)We suggest that the prior parameters (119886 119887) = (6 15) and(2 5) are appropriate for the square-root inverted-gammadistribution based on the Bayesian estimators under theRayleigh distribution

Acknowledgments

The authors thank the referees for their suggestions andhelpful comments on revising the paper This research waspartially supported by the National Science Council China(Plans nos NSC 101-2221-E-158-003 NSC 99-2118-M-415-001 and NSC 100-2118-M-415-002) Moreover J W Wu andC W Hong have a direct financial relation with the NationalScience Council Taiwan but other authors do not a directfinancial relation with the National Science Council Taiwan

References

[1] D C Montgomery Introduction To Statistical Quality ControlJohn Wiley amp Sons New York NY USA 1985

[2] L I Tong K S Chen and H T Chen ldquoStatistical testingfor assessing the performance of lifetime index of electroniccomponents with exponential distributionrdquo Journal of QualityReliability Management vol 19 pp 812ndash824 2002

[3] C-W Hong J-W Wu and C-H Cheng ldquoComputationalprocedure of performance assessment of lifetime index of


businesses for the Pareto lifetime model with the right type IIcensored samplerdquo Applied Mathematics and Computation vol184 no 2 pp 336ndash350 2007

[4] C W Hong J W Wu and C H Cheng ldquoComputational pro-cedure of performance assessment of lifetime index of Paretolifetime businesses based on confidence intervalrdquo Applied SoftComputing vol 8 no 1 pp 698ndash705 2008

[5] J-W Wu H-M Lee and C-L Lei ldquoComputational testingalgorithmic procedure of assessment for lifetime performanceindex of products with two-parameter exponential distribu-tionrdquo Applied Mathematics and Computation vol 190 no 1 pp116ndash125 2007

[6] W-C Lee J-W Wu and C-W Hong ldquoAssessing the lifetimeperformance index of products with the exponential distribu-tion under progressively type II right censored samplesrdquo Journalof Computational and Applied Mathematics vol 231 no 2 pp648ndash656 2009

[7] W C Lee J W Wu and C L Lei ldquoEvaluating the lifetimeperformance index for the exponential lifetime productsrdquoApplied Mathematical Modelling vol 34 no 5 pp 1217ndash12242010

[8] K N Chandler ldquoThe distribution and frequency of recordvaluesrdquo Journal of the Royal Statistical Society B vol 14 pp 220ndash228 1952

[9] S K Bhattacharya and R K Tyagi ldquoBayesian survival analysisbased on the rayleigh modelrdquo Trabajos de Estadistica vol 5 no1 pp 81ndash92 1990

[10] A D Cliff and J K Ord ldquoModel building and the analysis ofspatial pattern in human geography (with discussion)rdquo Journalof the Royal Statistical Society B vol 37 no 3 pp 297ndash348 1975

[11] D D Dyer and C W Whisenand ldquoBest linear unbiasedestimator of the parameter of the Rayleigh distribution I Smallsample theory for censored order statisticsrdquo IEEE Transactionson Reliability vol R-22 no 1 pp 27ndash34 1973

[12] A A Soliman and F M Al-Aboud ldquoBayesian inference usingrecord values from Rayleigh model with applicationrdquo EuropeanJournal of Operational Research vol 185 no 2 pp 659ndash6722008

[13] StatSoft STATISTICA Software Version 7 StatSoft Tulsa Okla2005

[14] W-C Lee J-W Wu M-L Hong L-S Lin and R-L ChanldquoAssessing the lifetime performance index of Rayleigh productsbased on the Bayesian estimation under progressive type IIright censored samplesrdquo Journal of Computational and AppliedMathematics vol 235 no 6 pp 1676ndash1688 2011

[15] M Ahsanullah Record Statistics Nova Science New York NYUSA 1995

[16] B C Arnold N Balakrishnan and H N Nagaraja RecordsJohn Wiley amp Sons New York NY USA 1998

[17] P W Zehna ldquoInvariance of maximum likelihood estimatorsrdquoAnnals of Mathematical Statistics vol 37 p 744 1966

[18] A Zellner ldquoBayesian estimation and prediction using asymmet-ric loss functionsrdquo Journal of the American Statistical Associa-tion vol 81 no 394 pp 446ndash451 1986

[19] S-Y Huang ldquoEmpirical Bayes testing procedures in somenonexponential families using asymmetric linex loss functionrdquoJournal of Statistical Planning and Inference vol 46 no 3 pp293ndash309 1995

[20] R Calabria andG Pulcini ldquoPoint estimation under asymmetricloss functions for left-truncated exponential samplesrdquo Commu-nications in StatisticsmdashTheory and Methods vol 25 no 3 pp585ndash600 1996

[21] A T K Wan G Zou and A H Lee ldquoMinimax and Γ-minimaxestimation for the Poisson distribution under LINEX loss whenthe parameter space is restrictedrdquo Statistics and ProbabilityLetters vol 50 no 1 pp 23ndash32 2000

[22] S Anatolyev ldquoKernel estimation under linear-exponential lossrdquoEconomics Letters vol 91 no 1 pp 39ndash43 2006

[23] X Li Y Shi J Wei and J Chai ldquoEmpirical Bayes estimators ofreliability performances using LINEX loss under progressivelytype-II censored samplesrdquo Mathematics and Computers inSimulation vol 73 no 5 pp 320ndash326 2007

[24] D K Dyer and P S L Liu ldquoOn comparison of estimators in ageneralized life modelrdquo Microelectronics Reliability vol 32 no1-2 pp 207ndash221 1992

[25] A A Soliman ldquoEstimation of parameters of life from progres-sively censored data using Burr-XII modelrdquo IEEE Transactionson Reliability vol 54 no 1 pp 34ndash42 2005

[26] A A Soliman and G R Elkahlout ldquoBayes estimation of thelogistic distribution based on progressively censored samplesrdquoJournal of Applied Statistical Science vol 14 no 3-4 pp 281ndash2932005

[27] C Caroni ldquoThe correct ldquoball bearingsrdquo datardquo Lifetime DataAnalysis vol 8 no 4 pp 395ndash399 2002

[28] M Z Raqab and M T Madi ldquoBayesian prediction of the totaltime on test using doubly censored Rayleigh datardquo Journal ofStatistical Computation and Simulation vol 72 no 10 pp 781ndash789 2002

[29] W C Lee ldquoStatistical testing for assessing lifetime performanceIndex of the Rayleigh lifetime productsrdquo Journal of the ChineseInstitute of Industrial Engineers vol 25 pp 433ndash445 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of


breaking stress or failure point varies even among identicalitems Hence in such experiments measurements may bemade sequentially and only the record values (lower or upper)are observed Record values arise naturally in many real-life applications involving data relating to weather sportseconomics and life tests According to themodel of Chandler[8] there are some situations in lifetime testing experimentswhere the failure time of a product is recorded if it exceedsall preceding failure times These recorded failure times arethe upper record value sequence In general let 119883

1 1198832

be a sequence of independent and identically distributed(iid) random variables having the same distribution asthe (population) random variable 119883 with the cumulativedistribution function (cdf) 119865(119909) and the probability densityfunction (pdf) 119891(119909) An observation 119883

119895will be called an

upper record value if it exceeds in value all of the precedingobservations that is if119883

119895gt 119883119894 for every 119894 lt 119895 The sequence

of record times 119879119899 119899 ge 1 is defined as follows

Let 1198791= 1 with probability 1 for 119899 ge 2 119879

119899= min119895

119883119895gt 119883119879119899minus1

A sequence of upper record values 119883119880(119899) is

then defined by

119883119880(119899)

= 119883119879119899

119899 = 1 2 (1)

Let 119883sim= 119883119880(1) 119883119880(2) 119883

119880(119899) be the first 119899 upper record

values arising from a sequence of iid random variables1198831 1198832 with cdf 119865(119909)

In this study we consider the case of the upper recordvalues instead of complete data or censored data In orderto evaluate the lifetime performance of nonnormal datawith upper record values the lifetime performance index 119862

119871

is utilized to measure product quality under the Rayleighdistribution with the upper record values The Rayleighdistribution is a nonnormal distribution and a special case ofthe Weibull distribution which provides a population modeluseful in several areas of statistics including life testing andreliability whose age with time as its failure rate is a linearfunction of time Bhattacharya and Tyagi [9] mentioned thatin some clinical studies dealing with cancer patients thesurvival pattern follows the Rayleigh distribution Cliff andOrd [10] showed that the Rayleigh distribution arises as thedistribution of the distance between an individual and itsnearest neighbor when the special pattern is generated by thePoisson process Dyer and Whisenand [11] demonstrated theimportance of this distribution in communication engineer-ing (also see Soliman and Al-Aboud [12])The pdf and cdfof the Rayleigh distribution are given respectively by

119891 (119909) =119909

1205792119890minus1199092

21205792

(2)

119865 (119909) = 1 minus 119890minus1199092

21205792

(3)

where 119909 gt 0 and 120579 gt 0 respectively When119883 comes from theRayleigh distribution the mean of 119883 is 119864(119883) = radic1205872120579 andthe standard deviation of119883 is 120590 = radic(4 minus 120587)2120579

Bayesian and non-Bayesian approaches have been used toobtain the estimators of the parameter 120579 under the Rayleigh

distribution with the upper record values Soliman and Al-Aboud [12] compared the performance of the Bayesian esti-mators with non-Bayesian estimators such as the MLE andthe best linear unbiased (BLUE) estimatorThe Bayesian esti-mators are developed under symmetric and nonsymmetricloss functions The symmetric loss function is squared-error(SE) loss function The nonsymmetric loss function includeslinear exponential (LINEX) and general entropy (GE) lossfunctions In recent decades the Bayesian viewpoint hasreceived frequent attention for analyzing failure data andother time-to-event data and has been often proposed as avalid alternative to traditional statistical perspectives TheBayesian approach to estimation of the parameters andreliability analysis allows prior subjective knowledge onlifetime parameters and technical information on the failuremechanism as well as experimental data Bayesian methodsusually require less sample data to achieve the same qualityof inferences than methods based on sampling theory (alsosee [12])

The main aim of this study will construct the Bayesianestimator of 119862


record values The Bayesian estimators of 119862119871are developed

under symmetric and nonsymmetric loss functions Theestimators of 119862

119871are then utilized to develop a credible

interval respectively The new testing procedures of credibleinterval for 119862

119871can be employed by managers to assess

whether the products performance adheres to the requiredlevel in the condition of known 119871 The new proposed testingprocedures can handle nonnormal lifetime data with upperrecord values Moreover we will evaluate the performance ofthe new proposed testing procedures with Bayesian and non-Bayesian approaches

The rest of this study is organized as follows Section 2introduces some properties of the lifetime performanceindex for lifetime of product with the Rayleigh distributionSection 3 discusses the relationship between the lifetimeperformance index and conforming rate Section 4 developsa hypothesis testing procedure for the lifetime performanceindex119862

119871with the non-Bayesian approach Section 5 develops

a hypothesis testing procedure for the lifetime performanceindex 119862

119871with the Bayesian approach Section 6 discusses the

MonteCarlo simulation algorithmof confidence (or credible)level One numerical example and concluding remarks aremade in Sections 7 and 8 respectively

2 The Lifetime Performance Index

Montgomery [1] has developed a process capability index 119862119871

to measure the larger-the-better quality characteristic Then119862119871is defined by

119862119871=120583 minus 119871

120590 (4)

where 120583 120590 and 119871 are the process mean the process standarddeviation and the lower specification limit respectively

To assess the product performance of products 119862119871can

be defined as the lifetime performance index If 119883 comesfrom theRayleigh distribution then the lifetime performance



119862119871=120583 minus 119871

120590=radic1205872120579 minus 119871


120587


2

4 minus 120587

119871

120579


120587

4 minus 120587

(5)





119875119903= 119875 (119883 ge 119871) = 119890



(6)




119871values and








between 119875119903and 119862



119903


119871with






0 where


0 119862119871le 1198880




= (119883119880(1) 119883119880(2) 119883




119880(119899)) is

119891 (119909119880(119899))

119899minus1

prod

119894=1

119891 (119909119880(119894))

1 minus 119865 (119909119880(119894)) (7)


119880(119899)) is given as

119871 (120579 | 119909119880(1) 119909

119880(119899)) =

119899

prod

119894=1

119909119880(119894)

1205792 exp(minus

1199092

119880(119899)

21205792) (8)


ln 119871 (120579 | 119909119880(1) 119909

119880(119899)) =

119899

sum

119894=1

ln (119909119880(119894)) minus 2119899 ln 120579 minus

1199092

119880(119899)

21205792

(9)


120579MLE =119883119880(119899)

radic2119899(10)



119862119871MLE = radic

120587


2

4 minus 120587

119871

120579MLE

= radic120587


2


radic2119899

119883119880(119899)

(11)




1198832




119880(119899)1205792sim 1205942

2119899 and 1205942

21198991minus120572


then

119875(1198832

119880(119899)

1205792le 1205942

21198991minus120572) = 1 minus 120572

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus119862119871MLE)(

1205942

21198991minus120572

2119899)

12

)

= 1 minus 120572

(12)




119903

119862119871

119875119903

119862119871

119875119903

119862119871

119875119903



119862119871ge radic

120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(13)



LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(14)





119871with


















119871with







0 where 119888



1 119862119871gt 1198880are





as follows

1205871 (120579) =

119886119887


119890minus1198862120579

2

(15)

where 120579 gt 0 119886 gt 0 and 119887 gt 0

119864 (120579) = (119886

2)

12Γ (119887 minus 12)

Γ (119887)

Var (120579) = 119886


119886

2) [Γ (119887 minus 12)

Γ (119887)]

2

119887 gt 1

(16)


119880(1) 119883119880(2) 119883



119883119880(2) 119883

119880(119899)) is given as

1199011(120579 | 119909

119880(1) 119909

119880(119899))

=119871 (120579 | 119909

119880(1) 119909

119880(119899)) 1205871 (120579)

intinfin

0119871 (120579 | 119909

119880(1) 119909

119880(119899)) 1205871 (120579) 119889120579

=

120579minus2(119899+119887)minus1

(1199092

119880(119899)+ 119886)(119899+119887)

119890minus(119909119880(119899)+119886)2120579

2

Γ (119899 + 119887) 2119899+119887minus1

(17)


119909119880(119899)) as (8)Let 119905 = 1199092


as

1199011(120579 | 119909

119880(1) 119909

119880(119899)) =

120579minus2(119899+119887)minus1

119905(119899+119887)

119890minus1199052120579

2

Γ (119899 + 119887) 2119899+119887minus1 (18)


=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)


119864 (120579 | 119909119880(1) 119909

119880(119899))

= int

infin

0

1205791199011(120579 | 119909

119880(1) 119909

119880(119899)) 119889120579

=Γ (119899 + 119887 minus 12)

Γ (119899 + 119887)(119905

2)

12

(19)

where 119905 = 1199092119880(119899)



120579BS =Γ (119899 + 119887 minus 12)

Γ (119899 + 119887)(119879

2)

12

(20)

where 119879 = 1198832




can be written as

119862119871= radic

120587


2

4 minus 120587

119871


119871lt radic

120587

4 minus 120587 (21)



119862119871BS = radic

120587


2

4 minus 120587

119871

120579BS

= radic120587


2


Γ (119899 + 119887 minus 12)

Γ (119899 + 119887)(119879

2)

12

]

minus1

(22)

where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the




=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

| 119909sim)

= 1 minus 120572

(23)



119871is given by

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

(24)



LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

(25)





119871with








sim= (119909

119880(1) 119909119880(2) 119909

119880(119899)) the













Δ1= (

120579lowast

120579)

2

minus 1 (27)




120579)2 | 119909sim]

minus 119888119864 [(120579lowast

120579)

2


(28)

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))



119889119864 [119871 (Δ1) | 119909sim]


120579)2 sdot2119888120579lowast

1205792| 119909sim]

minus 119888119864(2120579lowast

1205792| 119909sim) = 0

(29)


119864[120579BL1205792119890119888(120579BL120579)2 | 119909

sim] = 119890119888119864(

120579BL1205792

| 119909sim) (30)


2120579BL (119899 + 119887) sdot119905119899+119887

(119905 minus 21198881205792

BL)119899+119887+1

= 119890119888120579BL

2 (119899 + 119887)

119905

where 119905 = 1199092119880(119899)

+ 119886

(31)


120579BL = [119879

2119888(1 minus 119890

minus119888(119899+119887+1))]

12

(32)

where 119879 = 1198832




119862119871BL = radic

120587


2

4 minus 120587

119871

120579BL

= radic120587


2

4 minus 120587119871[119879

2119888(1 minus 119890

minus119888(119899+119887+1))]

minus12

(33)

where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the





=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

| 119909sim)

= 1 minus 120572

(34)



119871is

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

(35)



LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

(36)




119871with



















119871 (120579lowast 120579) prop (

120579lowast

120579)

119902


120579) minus 1 (37)



120579BG = [119864 (120579minus119902| 119909119880(1) 119909

119880(119899))]minus1119902

(38)


119864 (120579minus119902| 119909119880(1) 119909

119880(119899))

= int

infin

0

120579minus1199021199011(120579 | 119909

119880(1) 119909

119880(119899)) 119889120579

= (119905

2)

minus1199022

sdotΓ (119899 + 119887 + 1199022)

Γ (119899 + 119887)

(39)

where 119905 = 1199092119880(119899)


120579BG = (119879

2)

12

[Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(40)

where 119879 = 1198832




119862119871BG = radic

120587


2

4 minus 120587

119871

120579BG

= radic120587


2

4 minus 120587sdot 119871

sdot [(119879

2)

minus1199022Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

1119902

(41)


where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the




=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times[Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

| 119909sim)

= 1 minus 120572

(42)



119871is

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(43)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(44)




119871with



















119871 We considered 120572 = 005




following steps




1 1198852 119885



1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(45)

(1198841 1198842 119884



LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(46)






Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (47)



(1100)sum100

119894=1(1 minus 120572)

119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2




Step 3




1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(48)

(1198841 1198842 119884






Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (49)



119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2










7 Numerical Example




119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119888 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

minus05 094357 (000386) 093760 (000386) 093925 (000395)5 05 094357 (000386) 093760 (000386) 093925 (000395)

15 094357 (000386) 093760 (000386) 093925 (000395)minus05 094244 (000359) 093704 (000359) 093664 (000462)

10 05 094244 (000359) 093704 (000359) 093664 (000462)15 094244 (000359) 093704 (000359) 093664 (000462)minus05 094096 (000346) 093619 (000426) 093635 (000467)

15 05 094096 (000346) 093619 (000426) 093635 (000467)15 094096 (000346) 093619 (000426) 093635 (000467)



119899 119902 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

3 094357 (000386) 093760 (000386) 093925 (000395)5 5 094357 (000386) 093760 (000386) 093925 (000395)

8 094357 (000386) 093760 (000386) 093925 (000395)3 094244 (000359) 093704 (000359) 093664 (000462)

10 5 094244 (000359) 093704 (000359) 093664 (000462)8 094244 (000359) 093704 (000359) 093664 (000462)3 094096 (000346) 093619 (000426) 093635 (000467)

15 5 094096 (000346) 093619 (000426) 093635 (000467)8 094096 (000346) 093619 (000426) 093635 (000467)



6780 6780 6780 6864 3300 6864 9864 12804 42122892 4560 5184 5556 17340 4848 1788 9312 54124152 5196 12792 8412 10512 10584 and 6888







119880(119894) 119894 = 1 5(= 119899) = 6780 6864 9864

12804 17340


10010

99

90807060504030

20

10

5

32

1

Xi


()

Shape 2

Scale 8002N 25

AD 0431P -value 057





1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

= radic120587


120587

4 minus 120587minus 126251254)(

1205942

10095

10)

12

= 103285


119871is








1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1246074685)


Γ (5 + 1001)(1205942

2(5+1001)095

2)

12

= 096984

(51)




LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

= radic120587


120587

4 minus 120587minus 1129561059)

times [1

2 (05)(1 minus 119890

minus05(5+1001+1)) 1205942

2(5+1001)095]

12

= 096984

(52)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Τ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1200432998)

times [Γ (5 + 1001 + 22)

Γ (5 + 1001)]

minus12

(1205942

2(5+1001)095

2)

12

= 096984

(53)





0 119862119871le 090






8 Conclusions
















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119862119871=120583 minus 119871

120590=radic1205872120579 minus 119871


120587


2

4 minus 120587

119871

120579


120587

4 minus 120587

(5)





119875119903= 119875 (119883 ge 119871) = 119890



(6)




119871values and








between 119875119903and 119862



119903


119871with






0 where


0 119862119871le 1198880




= (119883119880(1) 119883119880(2) 119883




119880(119899)) is

119891 (119909119880(119899))

119899minus1

prod

119894=1

119891 (119909119880(119894))

1 minus 119865 (119909119880(119894)) (7)


119880(119899)) is given as

119871 (120579 | 119909119880(1) 119909

119880(119899)) =

119899

prod

119894=1

119909119880(119894)

1205792 exp(minus

1199092

119880(119899)

21205792) (8)


ln 119871 (120579 | 119909119880(1) 119909

119880(119899)) =

119899

sum

119894=1

ln (119909119880(119894)) minus 2119899 ln 120579 minus

1199092

119880(119899)

21205792

(9)


120579MLE =119883119880(119899)

radic2119899(10)



119862119871MLE = radic

120587


2

4 minus 120587

119871

120579MLE

= radic120587


2


radic2119899

119883119880(119899)

(11)




1198832




119880(119899)1205792sim 1205942

2119899 and 1205942

21198991minus120572


then

119875(1198832

119880(119899)

1205792le 1205942

21198991minus120572) = 1 minus 120572

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus119862119871MLE)(

1205942

21198991minus120572

2119899)

12

)

= 1 minus 120572

(12)




119903

119862119871

119875119903

119862119871

119875119903

119862119871

119875119903



119862119871ge radic

120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(13)



LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(14)





119871with


















119871with







0 where 119888



1 119862119871gt 1198880are





as follows

1205871 (120579) =

119886119887


119890minus1198862120579

2

(15)

where 120579 gt 0 119886 gt 0 and 119887 gt 0

119864 (120579) = (119886

2)

12Γ (119887 minus 12)

Γ (119887)

Var (120579) = 119886


119886

2) [Γ (119887 minus 12)

Γ (119887)]

2

119887 gt 1

(16)


119880(1) 119883119880(2) 119883



119883119880(2) 119883

119880(119899)) is given as

1199011(120579 | 119909

119880(1) 119909

119880(119899))

=119871 (120579 | 119909

119880(1) 119909

119880(119899)) 1205871 (120579)

intinfin

0119871 (120579 | 119909

119880(1) 119909

119880(119899)) 1205871 (120579) 119889120579

=

120579minus2(119899+119887)minus1

(1199092

119880(119899)+ 119886)(119899+119887)

119890minus(119909119880(119899)+119886)2120579

2

Γ (119899 + 119887) 2119899+119887minus1

(17)


119909119880(119899)) as (8)Let 119905 = 1199092


as

1199011(120579 | 119909

119880(1) 119909

119880(119899)) =

120579minus2(119899+119887)minus1

119905(119899+119887)

119890minus1199052120579

2

Γ (119899 + 119887) 2119899+119887minus1 (18)


=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)


119864 (120579 | 119909119880(1) 119909

119880(119899))

= int

infin

0

1205791199011(120579 | 119909

119880(1) 119909

119880(119899)) 119889120579

=Γ (119899 + 119887 minus 12)

Γ (119899 + 119887)(119905

2)

12

(19)

where 119905 = 1199092119880(119899)



120579BS =Γ (119899 + 119887 minus 12)

Γ (119899 + 119887)(119879

2)

12

(20)

where 119879 = 1198832




can be written as

119862119871= radic

120587


2

4 minus 120587

119871


119871lt radic

120587

4 minus 120587 (21)



119862119871BS = radic

120587


2

4 minus 120587

119871

120579BS

= radic120587


2


Γ (119899 + 119887 minus 12)

Γ (119899 + 119887)(119879

2)

12

]

minus1

(22)

where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the




=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

| 119909sim)

= 1 minus 120572

(23)



119871is given by

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

(24)



LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

(25)





119871with








sim= (119909

119880(1) 119909119880(2) 119909

119880(119899)) the













Δ1= (

120579lowast

120579)

2

minus 1 (27)




120579)2 | 119909sim]

minus 119888119864 [(120579lowast

120579)

2


(28)

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))



119889119864 [119871 (Δ1) | 119909sim]


120579)2 sdot2119888120579lowast

1205792| 119909sim]

minus 119888119864(2120579lowast

1205792| 119909sim) = 0

(29)


119864[120579BL1205792119890119888(120579BL120579)2 | 119909

sim] = 119890119888119864(

120579BL1205792

| 119909sim) (30)


2120579BL (119899 + 119887) sdot119905119899+119887

(119905 minus 21198881205792

BL)119899+119887+1

= 119890119888120579BL

2 (119899 + 119887)

119905

where 119905 = 1199092119880(119899)

+ 119886

(31)


120579BL = [119879

2119888(1 minus 119890

minus119888(119899+119887+1))]

12

(32)

where 119879 = 1198832




119862119871BL = radic

120587


2

4 minus 120587

119871

120579BL

= radic120587


2

4 minus 120587119871[119879

2119888(1 minus 119890

minus119888(119899+119887+1))]

minus12

(33)

where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the





=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

| 119909sim)

= 1 minus 120572

(34)



119871is

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

(35)



LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

(36)




119871with



















119871 (120579lowast 120579) prop (

120579lowast

120579)

119902


120579) minus 1 (37)



120579BG = [119864 (120579minus119902| 119909119880(1) 119909

119880(119899))]minus1119902

(38)


119864 (120579minus119902| 119909119880(1) 119909

119880(119899))

= int

infin

0

120579minus1199021199011(120579 | 119909

119880(1) 119909

119880(119899)) 119889120579

= (119905

2)

minus1199022

sdotΓ (119899 + 119887 + 1199022)

Γ (119899 + 119887)

(39)

where 119905 = 1199092119880(119899)


120579BG = (119879

2)

12

[Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(40)

where 119879 = 1198832




119862119871BG = radic

120587


2

4 minus 120587

119871

120579BG

= radic120587


2

4 minus 120587sdot 119871

sdot [(119879

2)

minus1199022Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

1119902

(41)


where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the




=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times[Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

| 119909sim)

= 1 minus 120572

(42)



119871is

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(43)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(44)




119871with



















119871 We considered 120572 = 005




following steps




1 1198852 119885



1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(45)

(1198841 1198842 119884



LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(46)






Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (47)



(1100)sum100

119894=1(1 minus 120572)

119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2




Step 3




1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(48)

(1198841 1198842 119884






Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (49)



119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2










7 Numerical Example




119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119888 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

minus05 094357 (000386) 093760 (000386) 093925 (000395)5 05 094357 (000386) 093760 (000386) 093925 (000395)

15 094357 (000386) 093760 (000386) 093925 (000395)minus05 094244 (000359) 093704 (000359) 093664 (000462)

10 05 094244 (000359) 093704 (000359) 093664 (000462)15 094244 (000359) 093704 (000359) 093664 (000462)minus05 094096 (000346) 093619 (000426) 093635 (000467)

15 05 094096 (000346) 093619 (000426) 093635 (000467)15 094096 (000346) 093619 (000426) 093635 (000467)



119899 119902 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

3 094357 (000386) 093760 (000386) 093925 (000395)5 5 094357 (000386) 093760 (000386) 093925 (000395)

8 094357 (000386) 093760 (000386) 093925 (000395)3 094244 (000359) 093704 (000359) 093664 (000462)

10 5 094244 (000359) 093704 (000359) 093664 (000462)8 094244 (000359) 093704 (000359) 093664 (000462)3 094096 (000346) 093619 (000426) 093635 (000467)

15 5 094096 (000346) 093619 (000426) 093635 (000467)8 094096 (000346) 093619 (000426) 093635 (000467)



6780 6780 6780 6864 3300 6864 9864 12804 42122892 4560 5184 5556 17340 4848 1788 9312 54124152 5196 12792 8412 10512 10584 and 6888







119880(119894) 119894 = 1 5(= 119899) = 6780 6864 9864

12804 17340


10010

99

90807060504030

20

10

5

32

1

Xi


()

Shape 2

Scale 8002N 25

AD 0431P -value 057





1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

= radic120587


120587

4 minus 120587minus 126251254)(

1205942

10095

10)

12

= 103285


119871is








1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1246074685)


Γ (5 + 1001)(1205942

2(5+1001)095

2)

12

= 096984

(51)




LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

= radic120587


120587

4 minus 120587minus 1129561059)

times [1

2 (05)(1 minus 119890

minus05(5+1001+1)) 1205942

2(5+1001)095]

12

= 096984

(52)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Τ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1200432998)

times [Γ (5 + 1001 + 22)

Γ (5 + 1001)]

minus12

(1205942

2(5+1001)095

2)

12

= 096984

(53)





0 119862119871le 090






8 Conclusions
















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119903

119862119871

119875119903

119862119871

119875119903

119862119871

119875119903



119862119871ge radic

120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(13)



LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(14)





119871with


















119871with







0 where 119888



1 119862119871gt 1198880are





as follows

1205871 (120579) =

119886119887


119890minus1198862120579

2

(15)

where 120579 gt 0 119886 gt 0 and 119887 gt 0

119864 (120579) = (119886

2)

12Γ (119887 minus 12)

Γ (119887)

Var (120579) = 119886


119886

2) [Γ (119887 minus 12)

Γ (119887)]

2

119887 gt 1

(16)


119880(1) 119883119880(2) 119883



119883119880(2) 119883

119880(119899)) is given as

1199011(120579 | 119909

119880(1) 119909

119880(119899))

=119871 (120579 | 119909

119880(1) 119909

119880(119899)) 1205871 (120579)

intinfin

0119871 (120579 | 119909

119880(1) 119909

119880(119899)) 1205871 (120579) 119889120579

=

120579minus2(119899+119887)minus1

(1199092

119880(119899)+ 119886)(119899+119887)

119890minus(119909119880(119899)+119886)2120579

2

Γ (119899 + 119887) 2119899+119887minus1

(17)


119909119880(119899)) as (8)Let 119905 = 1199092


as

1199011(120579 | 119909

119880(1) 119909

119880(119899)) =

120579minus2(119899+119887)minus1

119905(119899+119887)

119890minus1199052120579

2

Γ (119899 + 119887) 2119899+119887minus1 (18)


=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)


119864 (120579 | 119909119880(1) 119909

119880(119899))

= int

infin

0

1205791199011(120579 | 119909

119880(1) 119909

119880(119899)) 119889120579

=Γ (119899 + 119887 minus 12)

Γ (119899 + 119887)(119905

2)

12

(19)

where 119905 = 1199092119880(119899)



120579BS =Γ (119899 + 119887 minus 12)

Γ (119899 + 119887)(119879

2)

12

(20)

where 119879 = 1198832




can be written as

119862119871= radic

120587


2

4 minus 120587

119871


119871lt radic

120587

4 minus 120587 (21)



119862119871BS = radic

120587


2

4 minus 120587

119871

120579BS

= radic120587


2


Γ (119899 + 119887 minus 12)

Γ (119899 + 119887)(119879

2)

12

]

minus1

(22)

where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the




=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

| 119909sim)

= 1 minus 120572

(23)



119871is given by

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

(24)



LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

(25)





119871with








sim= (119909

119880(1) 119909119880(2) 119909

119880(119899)) the













Δ1= (

120579lowast

120579)

2

minus 1 (27)




120579)2 | 119909sim]

minus 119888119864 [(120579lowast

120579)

2


(28)

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))



119889119864 [119871 (Δ1) | 119909sim]


120579)2 sdot2119888120579lowast

1205792| 119909sim]

minus 119888119864(2120579lowast

1205792| 119909sim) = 0

(29)


119864[120579BL1205792119890119888(120579BL120579)2 | 119909

sim] = 119890119888119864(

120579BL1205792

| 119909sim) (30)


2120579BL (119899 + 119887) sdot119905119899+119887

(119905 minus 21198881205792

BL)119899+119887+1

= 119890119888120579BL

2 (119899 + 119887)

119905

where 119905 = 1199092119880(119899)

+ 119886

(31)


120579BL = [119879

2119888(1 minus 119890

minus119888(119899+119887+1))]

12

(32)

where 119879 = 1198832




119862119871BL = radic

120587


2

4 minus 120587

119871

120579BL

= radic120587


2

4 minus 120587119871[119879

2119888(1 minus 119890

minus119888(119899+119887+1))]

minus12

(33)

where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the





=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

| 119909sim)

= 1 minus 120572

(34)



119871is

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

(35)



LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

(36)




119871with



















119871 (120579lowast 120579) prop (

120579lowast

120579)

119902


120579) minus 1 (37)



120579BG = [119864 (120579minus119902| 119909119880(1) 119909

119880(119899))]minus1119902

(38)


119864 (120579minus119902| 119909119880(1) 119909

119880(119899))

= int

infin

0

120579minus1199021199011(120579 | 119909

119880(1) 119909

119880(119899)) 119889120579

= (119905

2)

minus1199022

sdotΓ (119899 + 119887 + 1199022)

Γ (119899 + 119887)

(39)

where 119905 = 1199092119880(119899)


120579BG = (119879

2)

12

[Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(40)

where 119879 = 1198832




119862119871BG = radic

120587


2

4 minus 120587

119871

120579BG

= radic120587


2

4 minus 120587sdot 119871

sdot [(119879

2)

minus1199022Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

1119902

(41)


where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the




=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times[Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

| 119909sim)

= 1 minus 120572

(42)



119871is

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(43)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(44)




119871with



















119871 We considered 120572 = 005




following steps




1 1198852 119885



1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(45)

(1198841 1198842 119884



LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(46)






Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (47)



(1100)sum100

119894=1(1 minus 120572)

119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2




Step 3




1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(48)

(1198841 1198842 119884






Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (49)



119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2










7 Numerical Example




119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119888 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

minus05 094357 (000386) 093760 (000386) 093925 (000395)5 05 094357 (000386) 093760 (000386) 093925 (000395)

15 094357 (000386) 093760 (000386) 093925 (000395)minus05 094244 (000359) 093704 (000359) 093664 (000462)

10 05 094244 (000359) 093704 (000359) 093664 (000462)15 094244 (000359) 093704 (000359) 093664 (000462)minus05 094096 (000346) 093619 (000426) 093635 (000467)

15 05 094096 (000346) 093619 (000426) 093635 (000467)15 094096 (000346) 093619 (000426) 093635 (000467)



119899 119902 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

3 094357 (000386) 093760 (000386) 093925 (000395)5 5 094357 (000386) 093760 (000386) 093925 (000395)

8 094357 (000386) 093760 (000386) 093925 (000395)3 094244 (000359) 093704 (000359) 093664 (000462)

10 5 094244 (000359) 093704 (000359) 093664 (000462)8 094244 (000359) 093704 (000359) 093664 (000462)3 094096 (000346) 093619 (000426) 093635 (000467)

15 5 094096 (000346) 093619 (000426) 093635 (000467)8 094096 (000346) 093619 (000426) 093635 (000467)



6780 6780 6780 6864 3300 6864 9864 12804 42122892 4560 5184 5556 17340 4848 1788 9312 54124152 5196 12792 8412 10512 10584 and 6888







119880(119894) 119894 = 1 5(= 119899) = 6780 6864 9864

12804 17340


10010

99

90807060504030

20

10

5

32

1

Xi


()

Shape 2

Scale 8002N 25

AD 0431P -value 057





1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

= radic120587


120587

4 minus 120587minus 126251254)(

1205942

10095

10)

12

= 103285


119871is








1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1246074685)


Γ (5 + 1001)(1205942

2(5+1001)095

2)

12

= 096984

(51)




LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

= radic120587


120587

4 minus 120587minus 1129561059)

times [1

2 (05)(1 minus 119890

minus05(5+1001+1)) 1205942

2(5+1001)095]

12

= 096984

(52)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Τ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1200432998)

times [Γ (5 + 1001 + 22)

Γ (5 + 1001)]

minus12

(1205942

2(5+1001)095

2)

12

= 096984

(53)





0 119862119871le 090






8 Conclusions
















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




as follows

1205871 (120579) =

119886119887


119890minus1198862120579

2

(15)

where 120579 gt 0 119886 gt 0 and 119887 gt 0

119864 (120579) = (119886

2)

12Γ (119887 minus 12)

Γ (119887)

Var (120579) = 119886


119886

2) [Γ (119887 minus 12)

Γ (119887)]

2

119887 gt 1

(16)


119880(1) 119883119880(2) 119883



119883119880(2) 119883

119880(119899)) is given as

1199011(120579 | 119909

119880(1) 119909

119880(119899))

=119871 (120579 | 119909

119880(1) 119909

119880(119899)) 1205871 (120579)

intinfin

0119871 (120579 | 119909

119880(1) 119909

119880(119899)) 1205871 (120579) 119889120579

=

120579minus2(119899+119887)minus1

(1199092

119880(119899)+ 119886)(119899+119887)

119890minus(119909119880(119899)+119886)2120579

2

Γ (119899 + 119887) 2119899+119887minus1

(17)


119909119880(119899)) as (8)Let 119905 = 1199092


as

1199011(120579 | 119909

119880(1) 119909

119880(119899)) =

120579minus2(119899+119887)minus1

119905(119899+119887)

119890minus1199052120579

2

Γ (119899 + 119887) 2119899+119887minus1 (18)


=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)


119864 (120579 | 119909119880(1) 119909

119880(119899))

= int

infin

0

1205791199011(120579 | 119909

119880(1) 119909

119880(119899)) 119889120579

=Γ (119899 + 119887 minus 12)

Γ (119899 + 119887)(119905

2)

12

(19)

where 119905 = 1199092119880(119899)



120579BS =Γ (119899 + 119887 minus 12)

Γ (119899 + 119887)(119879

2)

12

(20)

where 119879 = 1198832




can be written as

119862119871= radic

120587


2

4 minus 120587

119871


119871lt radic

120587

4 minus 120587 (21)



119862119871BS = radic

120587


2

4 minus 120587

119871

120579BS

= radic120587


2


Γ (119899 + 119887 minus 12)

Γ (119899 + 119887)(119879

2)

12

]

minus1

(22)

where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the




=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

| 119909sim)

= 1 minus 120572

(23)



119871is given by

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

(24)



LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

(25)





119871with








sim= (119909

119880(1) 119909119880(2) 119909

119880(119899)) the













Δ1= (

120579lowast

120579)

2

minus 1 (27)




120579)2 | 119909sim]

minus 119888119864 [(120579lowast

120579)

2


(28)

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))



119889119864 [119871 (Δ1) | 119909sim]


120579)2 sdot2119888120579lowast

1205792| 119909sim]

minus 119888119864(2120579lowast

1205792| 119909sim) = 0

(29)


119864[120579BL1205792119890119888(120579BL120579)2 | 119909

sim] = 119890119888119864(

120579BL1205792

| 119909sim) (30)


2120579BL (119899 + 119887) sdot119905119899+119887

(119905 minus 21198881205792

BL)119899+119887+1

= 119890119888120579BL

2 (119899 + 119887)

119905

where 119905 = 1199092119880(119899)

+ 119886

(31)


120579BL = [119879

2119888(1 minus 119890

minus119888(119899+119887+1))]

12

(32)

where 119879 = 1198832




119862119871BL = radic

120587


2

4 minus 120587

119871

120579BL

= radic120587


2

4 minus 120587119871[119879

2119888(1 minus 119890

minus119888(119899+119887+1))]

minus12

(33)

where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the





=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

| 119909sim)

= 1 minus 120572

(34)



119871is

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

(35)



LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

(36)




119871with



















119871 (120579lowast 120579) prop (

120579lowast

120579)

119902


120579) minus 1 (37)



120579BG = [119864 (120579minus119902| 119909119880(1) 119909

119880(119899))]minus1119902

(38)


119864 (120579minus119902| 119909119880(1) 119909

119880(119899))

= int

infin

0

120579minus1199021199011(120579 | 119909

119880(1) 119909

119880(119899)) 119889120579

= (119905

2)

minus1199022

sdotΓ (119899 + 119887 + 1199022)

Γ (119899 + 119887)

(39)

where 119905 = 1199092119880(119899)


120579BG = (119879

2)

12

[Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(40)

where 119879 = 1198832




119862119871BG = radic

120587


2

4 minus 120587

119871

120579BG

= radic120587


2

4 minus 120587sdot 119871

sdot [(119879

2)

minus1199022Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

1119902

(41)


where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the




=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times[Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

| 119909sim)

= 1 minus 120572

(42)



119871is

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(43)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(44)




119871with



















119871 We considered 120572 = 005




following steps




1 1198852 119885



1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(45)

(1198841 1198842 119884



LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(46)






Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (47)



(1100)sum100

119894=1(1 minus 120572)

119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2




Step 3




1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(48)

(1198841 1198842 119884






Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (49)



119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2










7 Numerical Example




119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119888 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

minus05 094357 (000386) 093760 (000386) 093925 (000395)5 05 094357 (000386) 093760 (000386) 093925 (000395)

15 094357 (000386) 093760 (000386) 093925 (000395)minus05 094244 (000359) 093704 (000359) 093664 (000462)

10 05 094244 (000359) 093704 (000359) 093664 (000462)15 094244 (000359) 093704 (000359) 093664 (000462)minus05 094096 (000346) 093619 (000426) 093635 (000467)

15 05 094096 (000346) 093619 (000426) 093635 (000467)15 094096 (000346) 093619 (000426) 093635 (000467)



119899 119902 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

3 094357 (000386) 093760 (000386) 093925 (000395)5 5 094357 (000386) 093760 (000386) 093925 (000395)

8 094357 (000386) 093760 (000386) 093925 (000395)3 094244 (000359) 093704 (000359) 093664 (000462)

10 5 094244 (000359) 093704 (000359) 093664 (000462)8 094244 (000359) 093704 (000359) 093664 (000462)3 094096 (000346) 093619 (000426) 093635 (000467)

15 5 094096 (000346) 093619 (000426) 093635 (000467)8 094096 (000346) 093619 (000426) 093635 (000467)



6780 6780 6780 6864 3300 6864 9864 12804 42122892 4560 5184 5556 17340 4848 1788 9312 54124152 5196 12792 8412 10512 10584 and 6888







119880(119894) 119894 = 1 5(= 119899) = 6780 6864 9864

12804 17340


10010

99

90807060504030

20

10

5

32

1

Xi


()

Shape 2

Scale 8002N 25

AD 0431P -value 057





1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

= radic120587


120587

4 minus 120587minus 126251254)(

1205942

10095

10)

12

= 103285


119871is








1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1246074685)


Γ (5 + 1001)(1205942

2(5+1001)095

2)

12

= 096984

(51)




LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

= radic120587


120587

4 minus 120587minus 1129561059)

times [1

2 (05)(1 minus 119890

minus05(5+1001+1)) 1205942

2(5+1001)095]

12

= 096984

(52)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Τ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1200432998)

times [Γ (5 + 1001 + 22)

Γ (5 + 1001)]

minus12

(1205942

2(5+1001)095

2)

12

= 096984

(53)





0 119862119871le 090






8 Conclusions
















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014







119871with








sim= (119909

119880(1) 119909119880(2) 119909

119880(119899)) the













Δ1= (

120579lowast

120579)

2

minus 1 (27)




120579)2 | 119909sim]

minus 119888119864 [(120579lowast

120579)

2


(28)

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))



119889119864 [119871 (Δ1) | 119909sim]


120579)2 sdot2119888120579lowast

1205792| 119909sim]

minus 119888119864(2120579lowast

1205792| 119909sim) = 0

(29)


119864[120579BL1205792119890119888(120579BL120579)2 | 119909

sim] = 119890119888119864(

120579BL1205792

| 119909sim) (30)


2120579BL (119899 + 119887) sdot119905119899+119887

(119905 minus 21198881205792

BL)119899+119887+1

= 119890119888120579BL

2 (119899 + 119887)

119905

where 119905 = 1199092119880(119899)

+ 119886

(31)


120579BL = [119879

2119888(1 minus 119890

minus119888(119899+119887+1))]

12

(32)

where 119879 = 1198832




119862119871BL = radic

120587


2

4 minus 120587

119871

120579BL

= radic120587


2

4 minus 120587119871[119879

2119888(1 minus 119890

minus119888(119899+119887+1))]

minus12

(33)

where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the





=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

| 119909sim)

= 1 minus 120572

(34)



119871is

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

(35)



LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

(36)




119871with



















119871 (120579lowast 120579) prop (

120579lowast

120579)

119902


120579) minus 1 (37)



120579BG = [119864 (120579minus119902| 119909119880(1) 119909

119880(119899))]minus1119902

(38)


119864 (120579minus119902| 119909119880(1) 119909

119880(119899))

= int

infin

0

120579minus1199021199011(120579 | 119909

119880(1) 119909

119880(119899)) 119889120579

= (119905

2)

minus1199022

sdotΓ (119899 + 119887 + 1199022)

Γ (119899 + 119887)

(39)

where 119905 = 1199092119880(119899)


120579BG = (119879

2)

12

[Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(40)

where 119879 = 1198832




119862119871BG = radic

120587


2

4 minus 120587

119871

120579BG

= radic120587


2

4 minus 120587sdot 119871

sdot [(119879

2)

minus1199022Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

1119902

(41)


where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the




=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times[Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

| 119909sim)

= 1 minus 120572

(42)



119871is

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(43)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(44)




119871with



















119871 We considered 120572 = 005




following steps




1 1198852 119885



1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(45)

(1198841 1198842 119884



LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(46)






Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (47)



(1100)sum100

119894=1(1 minus 120572)

119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2




Step 3




1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(48)

(1198841 1198842 119884






Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (49)



119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2










7 Numerical Example




119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119888 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

minus05 094357 (000386) 093760 (000386) 093925 (000395)5 05 094357 (000386) 093760 (000386) 093925 (000395)

15 094357 (000386) 093760 (000386) 093925 (000395)minus05 094244 (000359) 093704 (000359) 093664 (000462)

10 05 094244 (000359) 093704 (000359) 093664 (000462)15 094244 (000359) 093704 (000359) 093664 (000462)minus05 094096 (000346) 093619 (000426) 093635 (000467)

15 05 094096 (000346) 093619 (000426) 093635 (000467)15 094096 (000346) 093619 (000426) 093635 (000467)



119899 119902 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

3 094357 (000386) 093760 (000386) 093925 (000395)5 5 094357 (000386) 093760 (000386) 093925 (000395)

8 094357 (000386) 093760 (000386) 093925 (000395)3 094244 (000359) 093704 (000359) 093664 (000462)

10 5 094244 (000359) 093704 (000359) 093664 (000462)8 094244 (000359) 093704 (000359) 093664 (000462)3 094096 (000346) 093619 (000426) 093635 (000467)

15 5 094096 (000346) 093619 (000426) 093635 (000467)8 094096 (000346) 093619 (000426) 093635 (000467)



6780 6780 6780 6864 3300 6864 9864 12804 42122892 4560 5184 5556 17340 4848 1788 9312 54124152 5196 12792 8412 10512 10584 and 6888







119880(119894) 119894 = 1 5(= 119899) = 6780 6864 9864

12804 17340


10010

99

90807060504030

20

10

5

32

1

Xi


()

Shape 2

Scale 8002N 25

AD 0431P -value 057





1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

= radic120587


120587

4 minus 120587minus 126251254)(

1205942

10095

10)

12

= 103285


119871is








1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1246074685)


Γ (5 + 1001)(1205942

2(5+1001)095

2)

12

= 096984

(51)




LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

= radic120587


120587

4 minus 120587minus 1129561059)

times [1

2 (05)(1 minus 119890

minus05(5+1001+1)) 1205942

2(5+1001)095]

12

= 096984

(52)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Τ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1200432998)

times [Γ (5 + 1001 + 22)

Γ (5 + 1001)]

minus12

(1205942

2(5+1001)095

2)

12

= 096984

(53)





0 119862119871le 090






8 Conclusions
















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

| 119909sim)

= 1 minus 120572

(34)



119871is

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

(35)



LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

(36)




119871with



















119871 (120579lowast 120579) prop (

120579lowast

120579)

119902


120579) minus 1 (37)



120579BG = [119864 (120579minus119902| 119909119880(1) 119909

119880(119899))]minus1119902

(38)


119864 (120579minus119902| 119909119880(1) 119909

119880(119899))

= int

infin

0

120579minus1199021199011(120579 | 119909

119880(1) 119909

119880(119899)) 119889120579

= (119905

2)

minus1199022

sdotΓ (119899 + 119887 + 1199022)

Γ (119899 + 119887)

(39)

where 119905 = 1199092119880(119899)


120579BG = (119879

2)

12

[Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(40)

where 119879 = 1198832




119862119871BG = radic

120587


2

4 minus 120587

119871

120579BG

= radic120587


2

4 minus 120587sdot 119871

sdot [(119879

2)

minus1199022Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

1119902

(41)


where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the




=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times[Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

| 119909sim)

= 1 minus 120572

(42)



119871is

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(43)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(44)




119871with



















119871 We considered 120572 = 005




following steps




1 1198852 119885



1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(45)

(1198841 1198842 119884



LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(46)






Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (47)



(1100)sum100

119894=1(1 minus 120572)

119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2




Step 3




1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(48)

(1198841 1198842 119884






Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (49)



119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2










7 Numerical Example




119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119888 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

minus05 094357 (000386) 093760 (000386) 093925 (000395)5 05 094357 (000386) 093760 (000386) 093925 (000395)

15 094357 (000386) 093760 (000386) 093925 (000395)minus05 094244 (000359) 093704 (000359) 093664 (000462)

10 05 094244 (000359) 093704 (000359) 093664 (000462)15 094244 (000359) 093704 (000359) 093664 (000462)minus05 094096 (000346) 093619 (000426) 093635 (000467)

15 05 094096 (000346) 093619 (000426) 093635 (000467)15 094096 (000346) 093619 (000426) 093635 (000467)



119899 119902 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

3 094357 (000386) 093760 (000386) 093925 (000395)5 5 094357 (000386) 093760 (000386) 093925 (000395)

8 094357 (000386) 093760 (000386) 093925 (000395)3 094244 (000359) 093704 (000359) 093664 (000462)

10 5 094244 (000359) 093704 (000359) 093664 (000462)8 094244 (000359) 093704 (000359) 093664 (000462)3 094096 (000346) 093619 (000426) 093635 (000467)

15 5 094096 (000346) 093619 (000426) 093635 (000467)8 094096 (000346) 093619 (000426) 093635 (000467)



6780 6780 6780 6864 3300 6864 9864 12804 42122892 4560 5184 5556 17340 4848 1788 9312 54124152 5196 12792 8412 10512 10584 and 6888







119880(119894) 119894 = 1 5(= 119899) = 6780 6864 9864

12804 17340


10010

99

90807060504030

20

10

5

32

1

Xi


()

Shape 2

Scale 8002N 25

AD 0431P -value 057





1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

= radic120587


120587

4 minus 120587minus 126251254)(

1205942

10095

10)

12

= 103285


119871is








1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1246074685)


Γ (5 + 1001)(1205942

2(5+1001)095

2)

12

= 096984

(51)




LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

= radic120587


120587

4 minus 120587minus 1129561059)

times [1

2 (05)(1 minus 119890

minus05(5+1001+1)) 1205942

2(5+1001)095]

12

= 096984

(52)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Τ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1200432998)

times [Γ (5 + 1001 + 22)

Γ (5 + 1001)]

minus12

(1205942

2(5+1001)095

2)

12

= 096984

(53)





0 119862119871le 090






8 Conclusions
















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




where 119879 = 1198832




119909sim

=(119909119880(1)119909119880(119899))


=(119909119880(1)119909119880(119899)) given the




=(119909119880(1)119909119880(119899))sim 1205942

2(119899+119887)

and 12059422(119899+119887)1minus120572


2(119899+119887) then

119875(119905

1205792le 1205942

2(119899+119887)1minus120572| 119909sim) = 1 minus 120572

where 119909sim= (119909119880(1) 119909119880(2) 119909

119880(119899))

997904rArr 119875(119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times[Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

| 119909sim)

= 1 minus 120572

(42)



119871is

119862119871ge radic

120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(43)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Γ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

(44)




119871with



















119871 We considered 120572 = 005




following steps




1 1198852 119885



1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(45)

(1198841 1198842 119884



LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

(46)






Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (47)



(1100)sum100

119894=1(1 minus 120572)

119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2




Step 3




1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(48)

(1198841 1198842 119884






Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (49)



119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2










7 Numerical Example




119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119888 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

minus05 094357 (000386) 093760 (000386) 093925 (000395)5 05 094357 (000386) 093760 (000386) 093925 (000395)

15 094357 (000386) 093760 (000386) 093925 (000395)minus05 094244 (000359) 093704 (000359) 093664 (000462)

10 05 094244 (000359) 093704 (000359) 093664 (000462)15 094244 (000359) 093704 (000359) 093664 (000462)minus05 094096 (000346) 093619 (000426) 093635 (000467)

15 05 094096 (000346) 093619 (000426) 093635 (000467)15 094096 (000346) 093619 (000426) 093635 (000467)



119899 119902 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

3 094357 (000386) 093760 (000386) 093925 (000395)5 5 094357 (000386) 093760 (000386) 093925 (000395)

8 094357 (000386) 093760 (000386) 093925 (000395)3 094244 (000359) 093704 (000359) 093664 (000462)

10 5 094244 (000359) 093704 (000359) 093664 (000462)8 094244 (000359) 093704 (000359) 093664 (000462)3 094096 (000346) 093619 (000426) 093635 (000467)

15 5 094096 (000346) 093619 (000426) 093635 (000467)8 094096 (000346) 093619 (000426) 093635 (000467)



6780 6780 6780 6864 3300 6864 9864 12804 42122892 4560 5184 5556 17340 4848 1788 9312 54124152 5196 12792 8412 10512 10584 and 6888







119880(119894) 119894 = 1 5(= 119899) = 6780 6864 9864

12804 17340


10010

99

90807060504030

20

10

5

32

1

Xi


()

Shape 2

Scale 8002N 25

AD 0431P -value 057





1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

= radic120587


120587

4 minus 120587minus 126251254)(

1205942

10095

10)

12

= 103285


119871is








1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1246074685)


Γ (5 + 1001)(1205942

2(5+1001)095

2)

12

= 096984

(51)




LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

= radic120587


120587

4 minus 120587minus 1129561059)

times [1

2 (05)(1 minus 119890

minus05(5+1001+1)) 1205942

2(5+1001)095]

12

= 096984

(52)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Τ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1200432998)

times [Γ (5 + 1001 + 22)

Γ (5 + 1001)]

minus12

(1205942

2(5+1001)095

2)

12

= 096984

(53)





0 119862119871le 090






8 Conclusions
















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014








Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (47)



(1100)sum100

119894=1(1 minus 120572)

119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2




Step 3




1= 212057921198851and1198842119894= 21205792119885119894+1198842

119894minus1 for 119894 = 2 119899

1198841= radic21205792119885

1 119884

119894= radic21205792119885

119894+ 1198842

119894minus1 for 119894 = 2 119899

(48)

(1198841 1198842 119884






Step 4


(total count)1000

Step 5


(1 minus 120572)1 (1 minus 120572)

2 (1 minus 120572)

100 (49)



119894


1 (1 minus 120572)

2 (1 minus 120572)

100SMSE = (1100)

times sum100

119894=1[(1 minus 120572)

119894minus (1 minus 120572)]

2










7 Numerical Example




119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119888 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

minus05 094357 (000386) 093760 (000386) 093925 (000395)5 05 094357 (000386) 093760 (000386) 093925 (000395)

15 094357 (000386) 093760 (000386) 093925 (000395)minus05 094244 (000359) 093704 (000359) 093664 (000462)

10 05 094244 (000359) 093704 (000359) 093664 (000462)15 094244 (000359) 093704 (000359) 093664 (000462)minus05 094096 (000346) 093619 (000426) 093635 (000467)

15 05 094096 (000346) 093619 (000426) 093635 (000467)15 094096 (000346) 093619 (000426) 093635 (000467)



119899 119902 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

3 094357 (000386) 093760 (000386) 093925 (000395)5 5 094357 (000386) 093760 (000386) 093925 (000395)

8 094357 (000386) 093760 (000386) 093925 (000395)3 094244 (000359) 093704 (000359) 093664 (000462)

10 5 094244 (000359) 093704 (000359) 093664 (000462)8 094244 (000359) 093704 (000359) 093664 (000462)3 094096 (000346) 093619 (000426) 093635 (000467)

15 5 094096 (000346) 093619 (000426) 093635 (000467)8 094096 (000346) 093619 (000426) 093635 (000467)



6780 6780 6780 6864 3300 6864 9864 12804 42122892 4560 5184 5556 17340 4848 1788 9312 54124152 5196 12792 8412 10512 10584 and 6888







119880(119894) 119894 = 1 5(= 119899) = 6780 6864 9864

12804 17340


10010

99

90807060504030

20

10

5

32

1

Xi


()

Shape 2

Scale 8002N 25

AD 0431P -value 057





1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

= radic120587


120587

4 minus 120587minus 126251254)(

1205942

10095

10)

12

= 103285


119871is








1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1246074685)


Γ (5 + 1001)(1205942

2(5+1001)095

2)

12

= 096984

(51)




LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

= radic120587


120587

4 minus 120587minus 1129561059)

times [1

2 (05)(1 minus 119890

minus05(5+1001+1)) 1205942

2(5+1001)095]

12

= 096984

(52)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Τ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1200432998)

times [Γ (5 + 1001 + 22)

Γ (5 + 1001)]

minus12

(1205942

2(5+1001)095

2)

12

= 096984

(53)





0 119862119871le 090






8 Conclusions
















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2



119899 119888 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

minus05 094357 (000386) 093760 (000386) 093925 (000395)5 05 094357 (000386) 093760 (000386) 093925 (000395)

15 094357 (000386) 093760 (000386) 093925 (000395)minus05 094244 (000359) 093704 (000359) 093664 (000462)

10 05 094244 (000359) 093704 (000359) 093664 (000462)15 094244 (000359) 093704 (000359) 093664 (000462)minus05 094096 (000346) 093619 (000426) 093635 (000467)

15 05 094096 (000346) 093619 (000426) 093635 (000467)15 094096 (000346) 093619 (000426) 093635 (000467)



119899 119902 119886 = 2 119887 = 5 119886 = 6 119887 = 15 119886 = 2 119887 = 2

3 094357 (000386) 093760 (000386) 093925 (000395)5 5 094357 (000386) 093760 (000386) 093925 (000395)

8 094357 (000386) 093760 (000386) 093925 (000395)3 094244 (000359) 093704 (000359) 093664 (000462)

10 5 094244 (000359) 093704 (000359) 093664 (000462)8 094244 (000359) 093704 (000359) 093664 (000462)3 094096 (000346) 093619 (000426) 093635 (000467)

15 5 094096 (000346) 093619 (000426) 093635 (000467)8 094096 (000346) 093619 (000426) 093635 (000467)



6780 6780 6780 6864 3300 6864 9864 12804 42122892 4560 5184 5556 17340 4848 1788 9312 54124152 5196 12792 8412 10512 10584 and 6888







119880(119894) 119894 = 1 5(= 119899) = 6780 6864 9864

12804 17340


10010

99

90807060504030

20

10

5

32

1

Xi


()

Shape 2

Scale 8002N 25

AD 0431P -value 057





1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

= radic120587


120587

4 minus 120587minus 126251254)(

1205942

10095

10)

12

= 103285


119871is








1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1246074685)


Γ (5 + 1001)(1205942

2(5+1001)095

2)

12

= 096984

(51)




LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

= radic120587


120587

4 minus 120587minus 1129561059)

times [1

2 (05)(1 minus 119890

minus05(5+1001+1)) 1205942

2(5+1001)095]

12

= 096984

(52)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Τ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1200432998)

times [Γ (5 + 1001 + 22)

Γ (5 + 1001)]

minus12

(1205942

2(5+1001)095

2)

12

= 096984

(53)





0 119862119871le 090






8 Conclusions
















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




10010

99

90807060504030

20

10

5

32

1

Xi


()

Shape 2

Scale 8002N 25

AD 0431P -value 057





1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBMLE = radic120587


120587

4 minus 120587minus 119862119871MLE)(

1205942

21198991minus120572

2119899)

12

= radic120587


120587

4 minus 120587minus 126251254)(

1205942

10095

10)

12

= 103285


119871is








1 2 3 4 5) = (6780 6864 9864 12804 17340)


119903of products




0 119862119871le 090 versus




119871 where

LBBS = radic120587


120587

4 minus 120587minus 119862119871BS)

timesΓ (119899 + 119887 minus 12)

Γ (119899 + 119887)(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1246074685)


Γ (5 + 1001)(1205942

2(5+1001)095

2)

12

= 096984

(51)




LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

= radic120587


120587

4 minus 120587minus 1129561059)

times [1

2 (05)(1 minus 119890

minus05(5+1001+1)) 1205942

2(5+1001)095]

12

= 096984

(52)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Τ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1200432998)

times [Γ (5 + 1001 + 22)

Γ (5 + 1001)]

minus12

(1205942

2(5+1001)095

2)

12

= 096984

(53)





0 119862119871le 090






8 Conclusions
















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






LBBL = radic120587


120587

4 minus 120587minus 119862119871BL)

times [1

2119888(1 minus 119890

minus119888(119899+119887+1)) 1205942

2(119899+119887)1minus120572]

12

= radic120587


120587

4 minus 120587minus 1129561059)

times [1

2 (05)(1 minus 119890

minus05(5+1001+1)) 1205942

2(5+1001)095]

12

= 096984

(52)



LBBG = radic120587


120587

4 minus 120587minus 119862119871BG)

times [Γ (119899 + 119887 + 1199022)

Τ (119899 + 119887)]

minus1119902

(1205942

2(119899+119887)1minus120572

2)

12

= radic120587


120587

4 minus 120587minus 1200432998)

times [Γ (5 + 1001 + 22)

Γ (5 + 1001)]

minus12

(1205942

2(5+1001)095

2)

12

= 096984

(53)





0 119862119871le 090






8 Conclusions
















Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014



evaluating the lifetime performance index based on the bayesian

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