sometimes pool t estimation – via shrinkage

Journal of Statistical Planning andInference 79 (1999) 191–204

www.elsevier.com/locate/jspi

Sometimes pool t estimation – via shrinkage

Rakesh Srivastava ∗, Zoola KapasiDepartment of Statistics, Saurashtra University, University Road, Rajkot-360 005, India

Received 27 May 1997; accepted 14 October 1998

Abstract

The performance of an estimator can be improved by incorporating some additional infor-mation(s) available besides the sample information. If two censored samples are available fromthe same exponential distribution, it is advantageous to pool the two samples for estimating themean life. Further, incorporating guess information facilitates accuracy borrowing by shrinkageto a guess point or interval. Both the views have been taken into consideration in the presentstudy. The present paper proposes an estimator for the mean life time of a two parameter expo-nential distribution, using conditional and=or guess information on it, when the two guaranteesare equal but unknown. The bias, mean square error and relative e�ciency of the proposedestimator have been studied. Some theoretical results have been derived. It is observed that theproposed testimator dominates the conventional estimator in certain range of life ratio, guess liferatio and shrinkage factor. Further, it is claimed that it always fares better than the preliminarytest estimator for mean life proposed by Gupta and Singh (Microelectron. Reliab., 1985, 25,881–887). c© 1999 Elsevier Science B.V. All rights reserved.

Keywords: Exponential distribution, Censored samples, Preliminary test, Conditional-guesstestimator, Bias, Mean-square error, Relative e�ciency, Signi�cance level, Shrinkage factor,Life-ratio, Guess life ratio

1. Introduction

The exponential distribution plays the same role in life testing experiment whichthe normal distribution does for agricultural and other experiments. A number of lifetest data have been examined (Davis, 1952) and it was observed that the exponentialdistribution �ts well in most of the cases. This distribution occurs in many contextssuch as waiting time problems, time intervals between mining accidents, the life span ofelectric bulbs, etc. (Maguire et al., 1952; Bartholomew, 1987; Epstein, 1958; Lawless,1983).

∗ Corresponding author.

0378-3758/99/$ - see front matter c© 1999 Elsevier Science B.V. All rights reserved.PII: S0378 -3758(98)00234 -1

192 R. Srivastava, Z. Kapasi / Journal of Statistical Planning and Inference 79 (1999) 191–204

Let us consider that we have two independent random samples of sizes n1 and n2available from two-parameter exponential distribution(s) de�ned by

f(xij;Ai; �i) =

(�i)−1 exp[− (xij − Ai)=�i]; xij ¿Ai; �i ¿ 0 (i = 1; 2);

0 otherwise;(1.1)

where Ai and �i (i = 1; 2) are the location and scale parameters, respectively. Theseare also interpreted as the minimum guarantee period, threshold, shift parameters orwithin which no failure can occur and mean life, respectively.Generally, the equipments produced are costly and life testing may be time consum-

ing or destructive by nature in such situations the experimenter resorts to censoring.Let us consider that r2 observations from a sample of n2 equipments produced beforemodi�cation(s) are available. A similar set of r1 observations from a sample of n1equipments produced after modi�cation(s) are also available. Here, the producer is in-terested in estimating the mean life of the equipment produced after modi�cation(s) inthe production procedure, hoping that the mean life will increase and attain a value ofthe desired speci�cation.Our object is to provide an estimator of �1 however, without loss of generality one

may provide an estimator of �2, using sample information available from two type IIcensored samples and utilizing additional information about the population parameters.Here, we have two uncertainties about �1. First, it is suspected that the mean life willincrease after some modi�cation(s) i.e. �1¿�2; where �2 is the mean life time withoutany modi�cation(s); further, it is suspected that after modi�cation(s) �1 may attain avalue �01 (say) which is the desired speci�cation i.e. �1¿�01. It implies that we havesome point guess �01 on �1.To resolve these uncertainties we apply preliminary tests of signi�cance (PTS). The

PT procedure is advantageous for life data problems were generally testing happens tobe destructive and expensive.The e�ect of using a PTS for subsequent estimation was �rst considered by Bancroft

(1944), who investigated the bias and mean-square error of variance estimator aftera PTS on the equality of two variances. Such a procedure of estimation has beenstudied by a number of authors in analysis of variance models, e.g. (Kitagawa, 1963;Bancroft, 1964; Singh and Gupta, 1976). Richards (1963) attempted to analyse theconsequences of using a PTS to determine whether to use a one parameter or twoparameter exponential distribution. Bancroft and Han (1977), and Han et al. (1988)have compiled a bibliography on inference procedures involving PTS.As we wish to incorporate the available information(s) on the basis of the outcome

of a PTS, the problem considered here has two separate tests for deciding whether touse or not to use the available information(s). This may be done by testing H0: �1=�2against H1: �1¿�2 and H′

0: �1 = �01 against H

′1: �1¿�01. Both the hypotheses may be

tested at some pre assigned level of signi�cance. However, we have taken the samelevel of signi�cance for both the hypotheses. Consequently, we may have followingpossibilities: accept H0 and H′

0; accept H0 and reject H′0; reject H0 and accept H

′0; reject

R. Srivastava, Z. Kapasi / Journal of Statistical Planning and Inference 79 (1999) 191–204 193

H0 and H′0. Finally, the proposed testimator may be constructed by utilizing one or

both the available information(s) depending upon the outcome of PTS. As the proposedestimator may incorporate the conditional or guess or both the information(s), basedon a test of signi�cance, the proposed estimator has been termed as CONDITIONAL– GUESS TESTIMATOR (�̂CG (say)).Further, the problem under consideration has the following three di�erent situations;

situation 1: A1 and A2 are known;situation 2: A1 = A2 but the common value is unknown;situation 3: Nothing is known about A1 and A2:The present paper deals with situation 2 only.Now, depending upon the outcome(s) of PTS as outlined earlier, the proposed esti-

mator includes alwayspool, neverpool, shrinkage estimator as its components. However,when we wish to combine both the information(s) i.e. conditional and guess, any suit-able weighted estimator can be thought of. An evident choice for proposing weightedestimators could be along the lines of “shrinkage technique” proposed by Thompson(1968).The statistics for testing the hypothesis H0 under various situations have been given

by Epstein and Tsao (1953). If �1 = �2, a MVU estimator of mean life for each ofthe situations using both the sets of observations has been given by Epstein and Sobel(1954). The MVU estimator of mean life on the basis of a single set of observationhas been given by Epstein and Sobel (1954). Gupta and Singh (1985) proposed apreliminary test estimator for life data. The present study is an attempt to combine thetwo lines of estimation procedure viz., estimation under conditional speci�cation andestimation utilizing a guess information.The plan of the paper is as follows: We outline our procedure for estimation in

Section 2, in Sections 3 and 4 we have obtained expressions for bias and mean-squareerror of the proposed testimator. In Section 5, we have established some theoreticalresults, and in Section 6, we have obtained relative e�ciency of the proposed testimatorwith neverpool estimator. Finally, Section 7 comprises of numerical computations andrecommendations regarding the application of the proposed testimator.

2. Conditional – guess testimator �̂CG of �1

Let x116x126 · · ·6x1r1 be the failure times of the �rst r1 items in a life test inwhich n1 items were placed on test. Further, let x216x226 · · ·6x2r2 be the failuretimes of the �rst r2 items in another life test in which n2 items were placed on test.Suppose that the underlying distribution of each xij (i = 1; 2; j = 1; 2; : : : ; ri) is twoparameter exponential as given in Eq. (1.1).We are interested in estimating �1, when it is suspected that �1¿�2 and �1¿�01. To

incorporate the available informations, we resort to the PTS and test the hypothesesH0: �1 = �2 against H1: �1¿�2 and H′

0: �1 = �01 against H

′1: �1¿�01 using the test

statistics proposed by Epstein and Tsao (1954) and Epstein and Sobel (1953).


For the situation under study there are two possibilities (i) x11¡x21 or (ii) x11¿x21.Thus, depending on the outcomes of PTS our testimator under (i) can be proposed asfollows:

�̂CG =

k[v1 + v2 + w2r1 + r2 − 1

]+ (1− k)�01 if H0 & H′

0 is accepted i:e:

f2¡�2 and T2¡�2;v1 + v2 + w2r1 + r2 − 1 if H0 is acc: & H′

0 is rej: i:e:

f2¡�2 and T2¡�2;

k[v1

r1 − 1]+ (1− k)�01 if H0 is rej: & H′

0 is acc: i:e:

f2¿�2 and T2¡�2;v1

r1 − 1 if H0 & H′0 are rejected i:e:

f2¿�2 and T2¿�2:

Further, for (ii), the estimator may be proposed as

�̂CG =

k[v1 + v2 + w1r1 + r2 − 1

]+ (1− k)�01 if H0 & H′

0 is accepted i:e:

f′2¡�′2 and T

′2¡�′2;

v1 + v2 + w1r1 + r2 − 1 if H0 is acc: & H′

0 is rej: i:e:

f′2¡�′2 and T

′2¿�′2;

k[v1 + w1r1

]+ (1− k)�01 if H0 is rej: & H′

0 is acc: i:e:

f′2¿�′2 and T

′2¡�′2;

v1 + w1r1

if H0 & H′0 are rejected i:e:

f′2¿�′2 and T

′2¿�′2;

where

f2 =r2

(r1 − 1)v1

v2 + w2; �2 = F(2r1 − 2; 2r2; �2);

f′2 =

(r2 − 1)r1

(v1 + w1)v2

; �′2 = F(2r1; 2r2 − 2; �′2);

T2 =2v1�01; �2 = �2(2r1 − 2; �2);

T ′2 =

2(v1 + w1)�01

; �′2 = �2(2r1; �′2); �2 = �′2;

vi =ri∑j=1(xij − xi1) + (ni − ri)(xiri − xi1); i = 1; 2;

w1 = n1(x11 − x21) for x11¿x21;

w2 = n2(x21 − x11) for x11¡x21;


k is the shrinkage factor lying between 0 and 1. Here, F(m; n; �) is the upper 100�%point of F-distribution with (m; n) degrees of freedom and �2(2r1; �) is the value ofcentral chi-square variate satisfying the following relation

�=∫ z

0f(�22r1 ) d(�

22r1 ) where z = �2(2r1; �);

where f(�22r1 ) is the probability density function of the chi-square variate with 2r1degrees of freedom.

3. The bias of �̂CG

The Bias of �̂CG is de�ned as

BIAS(�̂CG) = E(�̂CG)− �1: (3.1)

In order to evaluate BIAS(�̂CG), �rst we have to �nd E(�̂CG).Now, the expected value of �̂CG is given by

E(�̂CG) = [E11P(A11; A13|D1) + E12P(A11; A′13|D1) + E13P(A12; A′13|D1)

+E14P(A12; A′13|D1)]P(D1)

+ [E21P(A21; A23|D2) + E22P(A21; A′23|D2) + E23P(A22; A23|D2)

+E24P(A22; A′23|D2)]P(D2); (3.2)

where

D1: x11¡x21; D2 : x11¿x21;

A11: f2¡�2; A21 : f′2¡�′2;

A12: f2¿�2; A22 : f′2¿�′2;

A13: T2¡�2; A23 : T ′2¡�′2;

A13: T2¿�2; A23 : T ′2¿�′2:

Terms Eij’s (i = 1; 2; j = 1; : : : ; 4) appearing in Eq. (3.2) represent the expectationsunder acceptances and=or rejections of H0 and H′

0. We have used several transforma-tions and some standard results of integrations to evaluate these expectations. Also,substituting the following results due to Epstein and Tsao (1953),

P(D1) = n1�=(n1�+ n2); (3.3)

P(D2) = n2=(n1�+ n2); (3.4)


in Eq. (3.2), �nally we get the expression for bias of �̂CG as a fraction of �1 as under

B1 =Bias(�̂CG)

�1;

B1 =n1�

(n1�+ n2)r2

(r1 + r2 − 1) [{�Ix2 (r1 − 1; r2 + 1)− Ix2 (r1; r2)}

×{1− (1− k)It(�1=2; r1 − 1)}] + n2(n1�+ n2)

(r2 − 1)(r1 + r2 − 1) [{�Ix

′2(r1; r2)

− Ix′2 (r1 + 1; r2 − 1)}{1− (1− k)It(�′2=2; r1)}]− [{(1− k)=(n1�+ n2)}×{1− �1}{n1�It(�2=2; r1 − 1) + n2It(�′2=2; r1)}]; (3.5)

where

�= �2=�1; �1 = �01=�1;

x2 = (r1 − 1)��2=[(r1 − 1)��2 + r2];x′2 = r1��

′2=[r1��

′2 + (r2 − 1)]:

As a partial check on Eq. (3.5), let �2=�′2=0 and �2=�′2=0, implying x2=x

′2=0. Then

we always reject the hypotheses H0 and H′0 and use neverpool estimator �̂1 de�ned as

follows:

�̂1 ={v1=(r1 − 1) if x11¡x21;(v1 + w1)=r1 if x11¿x21;

in this case B1 = 0.If we take the limits �2 and �′2 tending to in�nity and �2 = �

′2 = 0 i.e. never reject

H0 and always reject H′0. In this situation we use the alwayspool estimator �̂12 de�ned

as follows:

�̂12 ={(v1 + v2 + w2)=(r1 + r2 − 1) if x11¡x21;(v1 + v2 + w1)=(r1 + r2 − 1) if x11¿x21:

Then,

B1 = C�1(n1�r2 + n2r2 − n2)(�− 1) = E(�̂12)− �1;where

C = [(n1�+ n2)(r1 + r2 − 1)]−1:

4. Mean-square error of �̂CG

In order to evaluate the mean square error of �̂CG we use the relation

MSE(�̂CG) = E(�̂2CG)− 2�1Bias(�̂CG)− �21: (4.1)

Now, we need an expression for E(�̂2CG). The term Bias(�̂CG) has already been eval-

uated in Section 3. For the evaluation of E(�̂2CG), we have followed the same method


used for the evaluation of E(�̂CG). After some simpli�cation we get the �nal expression

for E(�̂2CG) as

E(�̂CG) =�21r1

(r1 − 1) +n1��21

(n1�+ n2)

[{r1(r1 − 1)

(r1 + r2 − 1)2 −r1

(r1 − 1)}

×{1− (1− k2)It(�2=2; r1 − 1)}Ix2 (r1 + 1; r2) +2�r2(r1 − 1)(r1 + r2 − 1)2

×{1− (1− k2)It(�2=2; r1 − 1)}Ix2 (r1; r2 + 1) +�2r2(r2 + 1)(r1 + r2 − 1)2

−{1− (1− k2)It(�2=2; r1 − 1)}Ix2 (r1 − 1; r2 + 2)−2k(1− k)�1r2(r1 + r2 − 1)

×It(�2=2; r1 − 1){Ix2 (r1; r2)− �Ix2 (r1 − 1; r2 + 1)}+r1(k2 − 1)(r1 − 1)

×It(�2=2; r1 − 1) + (1− k)�1It(�2=2; r1 − 1){(1− k)�1 + 2k}]

+n2�21

(n1�+ n2)

[{r1(r1 + 1)

(r1 + r2 − 1)2 −(r1 + 1)r1

}Ix′2 (r1 + 2; r2 − 1)

×{1− (1− k2)Ii(�′2=2; r1)}+2�r1(r2 − 1)(r1 + r2 − 1)2 Ix

′2(r1 + 1; r2)

×{1− (1− k2)Ii(�′2=2; r1)}+�2r2(r2 − 1)(r1 + r2 − 1)2 Ix

′2(r1; r2 + 1)

×{1− (1− k2)Ii(�′2=2; r1)} −2k(1− k)�1(r2 − 1)

(r1 + r2 − 1) Ii(�′2=2; r1)

×{Ix′2 (r1 + 1; r2 − 1)− �Ix′2 (r1; r2)}+(k2 − 1)(r1 + 1)

r1Ii(�′2=2; r1)

− 1r1(r1 − 1) + (1− k)�1Ii(�

′2=2; r1){(1− k)�1 + 2k}): (4.2)

Substituting the values of E(�̂2CG) and Bias(�̂CG) from Eqs. (4.2) and (3.5), in Eq.

(4.1) and simplifying the results we obtain the expression of MSE(�̂CG) expressed asa fraction of �21 as follows:

M1 =MSE(�̂CG)

�21;

M1 =1

(r1 − 1) +n1�

(n1�+ n2)

({r1(r1 − 1)

(r1 + r2 − 1)2 −r1

(r1 − 1)}Ix2 (r1 + 1; r2)

×{1− (1− k2)Ii(�2=2; r1 − 1)}+ 2�r2(r1 − 1)(r1 + r2 − 1)2


×{1− (1− k2)Ii(�2=2; r1 − 1)}Ix2 (r1; r2 + 1) +�2r2(r2 + 1)(r1 + r2 − 1)2

×{1− (1− k2)Ii(�2=2; r1 − 1)}Ix2 (r1 − 1; r2 + 2)

+2r2

(r1 + r2 − 1){1− (1− k)(1 + k�1)Ii(�2=2; r1 − 1)}

×{Ix2 (r1; r2)−�Ix2 (r1−1; r2 + 1)}+(1− k)(r1 − 1){(1−k)r1−2}Ii(�2=2; r1−1)

+ (1− k)2�1(�1 − 2)Ii(�2=2; r1 − 1))

+n2

(n1�+ n2)

({r1(r1 + 1)

(r1 + r2 − 1)2 −(r1 + 1)r1

}{1− (1− k2)Ii(�′2=2; r1)}

×Ix′2 (r1 + 2; r2 − 1) +2�r1(r2 − 1)(r1 + r2 − 1)2 {1− (1− k

2)Ii(�′2=2; r1)}Ix′2 (r1 + 1; r2)

+�2r2(r2 − 1)(r1 + r2 − 1)2 {1− (1− k

2)Ii(�′2=2; r1)}Ix′2 (r1; r2 + 1)

+(1− k)r1

{(r1 − 1)− k(r1 + 1)}Ii(�′2=2; r1)

+2(r2 − 1)

(r1 + r2 − 1){1− (1− k)(1 + k�1)Ii(�′2=2; r1)}

×{Ix′2 (r1 + 1; r2 − 1)− �Ix′2 (r1; r2)}

+ (1− k)2�1(�1 − 2)Ii(�′2=2; r1)−1

r1(r1 − 1)): (4.3)

As a partial check on Eq. (4.3), let �2 = �′2 = 0 and �2 = �′2 = 0. Then Eq. (4.3)

reduces to

M1 =1

(r1 − 1)(1− n2

r1(n1�+ n2)

)=MSE(�̂1)=�21:

If we take �2 = �′2 → ∞ and �2 = �′2 = 0, then

M1 = l2l3[n1�{r1(r1 − 1) + 2�(r1 − 1)r2 + �2r2(r2 + 1)}+ n2{r1(r1 + 1)+2�r1(r2 − 1) + �2r2(r2 − 1)}]− 2(�− 1)l1{n1�r2 + n2r2 − n2} − 1;

M1 = MSE(�̂12)=�21;

where

l1 = [(n1�+ n2)(r1 + r2 − 1)]−1;l2 = [(n1�+ n2)]−1;l3 = (r1 + r2 − 1)2;

which is the same as obtained by Gupta and Singh (1985).


5. Theoretical result on MSE

Theorem. For 06k ¡ 1; the mean-square error of the conditional – guess testimatoris always less than that of the sometimes pool estimator.

Proof. Mean-square error of the sometimes pool estimator proposed by Gupta andSingh (1985) expressed as a fraction of �21 is given by

MSE(�̂1)/�̂21 =D1 + n1�l2({l3r1(r1−1)−D1r1}Ix2 (r1 + 1; r2) + 2�l3r2(r2−1)

×Ix2 (r1; r2 + 1) + �2l3r2(r2 + 1)Ix2 (r1 − 1; r2 + 2)) + n2l2({l3r1×(r1 + 1)− (r1 + 1)=r1}Ix′2 (r1 + 2; r2 − 1) + 2�l3r1(r2 − 1)×Ix′2 (r1 + 1; r2) + �2l3r2(r2 − 1)Ix′2 (r1; r2 + 1)− D1=r1)− 2B1;

(5.1)

where D1 = 1=(r1 − 1).Now, MSE of the proposed testimator derived in Section 4 expressed as a fraction

of �21 is given by Eq. (4.3).Taking di�erence of Eqs. (4.3) and (5.1), we observe that

MSE(�̂CG)�21

− MSE(�̂)�21

¡ 0;

which proves the result.

When k = 1, expression (4.3) reduces to Eq. (5.1) and hence in that case both theestimators have same mean-square error.

6. E�ciency of �̂CG

As preliminary test estimators are, in general, biased and neverpool estimators arealways unbiased and since the two are competing estimators of �1, it is more appropriateto talk of the relative e�ciency of the proposed testimator to the neverpool estimator.We now de�ne

RE =MSE (neverpool estimator)

MSE (conditional− guess testimator) ;

RE={

1(r1 − 1) +

n1�(n1�+ n2)

({r1(r1 − 1)

(r1 + r2 − 1)2 −r1

(r1 − 1)}Ix2 (r1 + 1; r2)

×{1− (1− k2)Ii(�2=2; r1 − 1)}+ 2�r2(r1 − 1)(r1 + r2 − 1)2

×{1− (1− k2)Ii(�2=2; r1 − 1)}Ix2 (r1; r2 + 1) +�2r2(r2 + 1)(r1 + r2 − 1)2


×{1− (1− k2)Ii(�2=2; r1 − 1)}Ix2 (r1 − 1; r2 + 2) +2r2

(r1 + r2 − 1)×{1− (1− k)(1 + k�1)Ii(�2=2; r1 − 1)}{Ix2 (r1; r2)− �Ix2 (r1 − 1; r2 + 1)}

+((1− k)r1 − 1 {(1− k)r1 − 2}Ii(�2=2; r1 − 1)

+ (1− k)2�1(�1 − 2)Ii(�2=2; r1 − 1))+

n2(n1�+ n2)

×({

r1(r1 + 1)(r1 + r2 − 1)2 −

(r1 + 1)r1

}{1−(1−k2)Ii(�′2=2; r1)}Ix′2 (r1 + 2; r2−1)

+2�r1(r2 − 1)(r1 + r2 − 1)2 {1− (1− k

2)Ii(�′2=2; r1)}Ix′2 (r1 + 1; r2) +�2r2(r2 − 1)(r1 + r2 − 1)2

×{1− (1− k2)Ii(�′2=2; r1)}Ix′2 (r1; r2 + 1) +(1− k)r1

×{(r1 − 1)− k(r1 + 1)}Ii(�′2=2; r1) +2(r2 − 1)

(r1 + r2 − 1)×{1− (1− k)(1 + k�1)Ii(�′2=2; r1)}{Ix′2 (r1 + 1; r2 − 1)− �Ix′2 (r1; r2)}

+ (1− k)2�1(�1 − 2)Ii(�′2=2; r1)−1

r1(r1 − 1))}−1

×(

1(r1 − 1)

{1− n2

r1(n1�+ n2)

}): (6.1)

From the above expression we observe that REF is a function of k; n1; n2; r1; r2; �; �1;�2; �′2; �2 and �

′2. Out of these parameters n1 and n2 are �xed in advance before per-

forming the experiment, r1 and r2 are determined and �xed by the experimenter takinginto consideration the cost and time involved in performing the experiment. The av-erage life ratios � and �1 are in general, unknown. k is the shrinkage factor lyingbetween 0 and 1. Hence, the only parameter(s) at our disposal is the level of signif-icance of PTS. For the sake of simplicity we have taken, without loss of generality,�2 = �′2 = �2 = �

′2 = �. However, one can take di�erent levels of signi�cance for both

the hypotheses, this could be a subject of future research.To study the behaviour of mean-square error of the proposed estimator �̂CG, we have

considered four sets of parameters given in Table 1.

7. Discussion on numerical results

As discussed earlier, relative e�ciency is a function of k; n1; n2; r1; r2; �; �1; �1 and�1. We have calculated the relative e�ciencies for the set-up given in Section 6. Itis to be noted that for each data set, we have taken four values of �, resulting in 16


Table 1Combinations of values of r1; r2; n1 and n2

Set no. n1 n2 r1 r2

1 30 30 5 92 30 30 5 133 50 50 17 94 50 50 17 13

Note: we have taken �=0:01; 0:05; 0:10; and 0.25; �=0:1(0:1)1:0; �1 =0:2(0:2)2:0; k = 0:1(0:1)1:0 for each r1; r2; n1 and n2.

Table 2E�ective ranges of �; �1 & k where RE¿ 1

Level(s) of signi�cance (�)

1% 5% 10% 25%

(1) For n1 = 30; n2 = 30; r1 = 5; r2 = 9� 0.5–1.0 0.5–1.0 0.5–1.0 0.5–1.0

(0.1–1.0) (0.1–1.0) (0.1–1.0) (0.1–1.0)�1 0.2–1.8 0.2–1.8 0.2–1.8 0.2–1.8

(0.6–1.4) (0.6–1.4) (0.6–1.4) (0.6–1.4)k 0.5–1.0 0.5–1.0 0.5–1.0 0.5–1.0

(0.1–1.0) (0.1–1.0) (0.1–1.0) (0.1–1.0)

(2) For n1 = 30; n2 = 30; r1 = 5; r2 = 13� 0.5–1.0 0.5–1.0 0.5–1.0 0.5–1.0

(0.1–1.0) (0.1–1.0) (0.1–1.0) (0.1–1.0)�1 0.2–1.8 0.2–1.8 0.2–1.8 0.2–1.8

(0.6–1.4) (0.6–1.4) (0.6–1.4) (0.6–1.4)k 0.5–1.0 0.5–1.0 0.5–1.0 0.5–1.0

(0.1–1.0) (0.1–1.0) (0.1–1.0) (0.1–1.0)

(3) For r1 = 50; r2 = 50; r1 = 17; r2 = 9� 0.5–1.0 0.5–1.0 0.5–1.0 0.5–1.0

(0.1–1.0) (0.1–1.0) (0.1–1.0) (0.1–1.0)�1 0.6–1.4 0.6–1.4 0.6–1.4 0.6–1.4

(0.8–1.2) (0.8–1.2) (0.8–1.2) (0.8–1.2)k 0.5–1.0 0.5–1.0 0.5–1.0 0.5–1.0

(0.1–1.0) (0.1–1.0) (0.1–1.0) (0.1–1.0)

(4) For n1 = 50; n2 = 50; r1 = 17; r2 = 13� 0.5–1.0 0.5–1.0 0.5–1.0 0.5–1.0

(0.1–1.0) (0.1–1.0) (0.1–1.0) (0.1–1.0)�1 0.6–1.4 0.6–1.4 0.6–1.4 0.6– 1.4

(0.8–1.2) (0.8–1.2) (0.8–1.2) (0.8–1.2)k 0.5–1.0 0.5–1.0 0.5–1.0 0.5–1.0

(0.1–1.0) (0.1–1.0) (0.1–1.0) (0.1–1.0)

tables for relative e�ciencies. However, due to space constraints all these tables havenot been presented here.A study of these tables reveals that the proposed testimator performs well for �=1%,

whatever may be the data set. We have therfore presented only four tables in theappendix (Tables 3–6). However, our �ndings based on Table 2 are as follows.


Table 3Relative e�ciency of �̂CG with respect to neverpool estimator for n1 = 30; n2 = 30; r1 = 5; r2 = 9; �= 1%

� k �1

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.1 0.1 0.40 0.70 1.54 5.62 48.99 5.66 1.55 0.70 0.40 0.250.2 0.2 0.49 0.86 1.80 5.46 18.23 5.88 1.89 0.89 0.51 0.330.3 0.3 0.61 1.02 1.97 4.58 8.92 5.28 2.22 1.21 0.66 0.430.4 0.4 0.76 1.20 2.06 3.74 5.45 4.31 2.41 1.37 0.85 0.570.5 0.5 0.96 1.40 2.10 3.12 3.84 3.42 2.40 1.58 1.07 0.750.6 0.6 1.22 1.63 2.15 2.68 2.94 2.74 2.24 1.71 1.27 0.960.7 0.7 1.53 1.83 2.11 2.31 2.37 2.24 2.00 1.70 1.41 1.160.8 0.8 1.76 1.87 1.95 1.97 1.94 1.86 1.74 1.60 1.45 1.300.9 0.9 1.67 1.66 1.64 1.62 1.58 1.54 1.50 1.45 1.39 1.341.0 1.0 1.27 1.27 1.27 1.27 1.27 1.27 1.27 1.27 1.27 1.27

Table 4Relative e�ciency of �̂CG with respect to neverpool estimator for n1 = 30; n2 = 30; r1 = 5; r2 = 13; �=1%

� k �1

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.1 0.1 0.40 0.70 1.54 5.63 49.34 5.66 1.55 0.70 0.40 0.250.2 0.2 0.49 0.86 1.80 5.45 18.06 5.87 1.89 0.88 0.51 0.330.3 0.3 0.60 1.00 1.92 4.40 8.52 5.24 2.23 1.13 0.66 0.430.4 0.4 0.73 1.14 1.94 3.45 5.11 4.25 2.45 1.40 0.87 0.580.5 0.5 0.89 1.30 1.94 2.86 3.61 3.39 2.47 1.65 1.11 0.780.6 0.6 1.12 1.49 1.98 2.50 2.84 2.77 2.33 1.80 1.35 1.020.7 0.7 1.42 1.71 2.01 2.26 2.37 2.32 2.11 1.82 1.53 1.260.8 0.8 1.73 1.87 1.97 2.03 2.04 1.99 1.88 1.74 1.59 1.430.9 0.9 1.78 1.79 1.78 1.76 1.74 1.70 1.65 1.60 1.55 1.481.0 1.0 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43 1.43

In Table 2, the values given in the parenthesis represent e�ective ranges of �; �1 andk, where relative e�ciency is greater than 1 but, here the e�ective ranges are wider for� and k. It is observed that, an increase in the e�ective range of �1, reduces e�ectiverange of (k; �) and vice versa.

8. Concluding remarks

A conditional-guess testimator has been proposed to re ect the idea of combiningconditional and=or guess information available from any reliable source(s). It has beenobserved that it fares better than the sometimes pool estimator almost always for a widerrange of life ratio, guess life ratio and shrinkage factor. Proposed testimator performsbetter than the neverpool estimator when k¿1− 2=r1. However, when k ¡ 1− 2=r1 itfares better than the neverpool estimator by taking large value of �1. These �ndingsare supported by numerical computations assembled in the appendix. Further, numericalcomputations suggest that, a lower level of signi�cance and a small censoring fractionis preferable.



� k �1

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.1 0.1 0.12 0.20 0.46 1.78 49.96 1.78 0.46 0.20 0.12 0.070.2 0.2 0.15 0.26 0.57 2.10 20.11 2.10 0.57 0.26 0.15 0.090.3 0.3 0.19 0.33 0.71 2.36 10.08 2.36 0.71 0.33 0.19 0.120.4 0.4 0.25 0.43 0.89 2.47 5.95 2.46 0.89 0.43 0.25 0.160.5 0.5 0.35 0.58 1.10 2.39 3.92 2.37 1.09 0.57 0.35 0.230.6 0.6 0.49 0.77 1.29 2.16 2.77 2.13 1.27 0.76 0.49 0.330.7 0.7 0.71 1.00 1.41 1.86 2.07 1.83 1.37 0.97 0.69 0.510.8 0.8 0.98 1.18 1.39 1.55 1.60 1.53 1.36 1.15 0.94 0.770.9 0.9 1.14 1.20 1.25 1.27 1.28 1.26 1.23 1.17 1.11 1.041.0 1.0 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04


� k �1

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.1 0.1 0.12 0.20 0.46 1.78 49.96 1.78 0.46 0.20 0.12 0.070.2 0.2 0.15 0.26 0.57 2.10 20.10 2.10 0.57 0.26 0.15 0.090.3 0.3 0.19 0.33 0.71 2.36 10.06 2.35 0.71 0.33 0.19 0.120.4 0.4 0.25 0.43 0.89 2.46 5.95 2.46 0.89 0.43 0.25 0.160.5 0.5 0.35 0.58 1.10 2.40 3.94 2.38 1.09 0.57 0.35 0.230.6 0.6 0.50 0.78 1.31 2.20 2.82 2.15 1.27 0.76 0.49 0.330.7 0.7 0.73 1.02 1.46 1.93 2.13 1.86 1.38 0.97 0.69 0.500.8 0.8 1.03 1.25 1.48 1.64 1.68 1.58 1.38 1.16 0.94 0.760.9 0.9 1.24 1.30 1.34 1.36 1.35 1.32 1.28 1.21 1.14 1.051.0 1.0 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11 1.11

Acknowledgements

The authors are thankful to the honourable referee for constructive suggestions andvaluable comments.

Appendix

The relative e�ciency of �̂CG with respect to neverpool estimator is shown in Tables3–6.

References

Bancroft, T.A., 1944. On biases in estimation due to the use of preliminary test of signi�cance. Ann. Math.Statist. 15, 17–32.

Bancroft, T.A., 1964. Analysis and inference for incompletely speci�ed models involving the use ofpreliminary test(s) of signi�cance. Biometrics 20, 427–442.


Bancroft, T.A., Han, C.P., 1977. Inference based on conditional speci�cation: A note and a bibliography.Int. Statist. Rev. 45, 117–127.

Bartholomew, D.J., 1987. A problem in life testing. J. Amer. Statist. Assoc. 52, 350–355.Davis, D.J., 1952. The analysis of some failure data. J. Amer. Statist. Assoc. 47, 113–160.Epstein, B., 1958. Exponential distribution and its role in life testing. Indian Quality Control 15, 4–6.Epstein, B., Sobel, M., 1954. Some theorems relevant to life testing from an exponential distribution. Ann.Math. Statist. 25, 373–381.

Epstein, B., Tsao, C.K., 1953. Some tests based on ordered observations from two exponential populations.Ann. Math. Statist. 24, 458–466.

Gupta, V.P., Singh, U., 1985. Preliminary test estimator for life data. Microelectron. Reliab. 25, 881–887.Han, C.P., Rao, C.V., Ravichandran, J., 1988. Inference based on conditional speci�cation: A secondbibliography. Comm. Statist. Theory Methods 17 (6), 1945–1964.

Kitagawa, T., 1963. Estimation after preliminary tests of signi�cance. Univ. California Publ. Statist. 3,147–186.

Lawless, J.F., 1983. Statistical methods in reliability. Technometrics 25, 305–316.Maguire, B.A., Pearson, E.S., Wynn, A.H.A., 1952. The time intervals between industrial accidents.Biometrika 39, 168–180.

Richards, D.O., 1963. Incompletely speci�ed models in life testing. Unpublished Ph.D. Thesis, Iowa StateUniversity.

Singh, U., Gupta, V.P., 1976. Bias and mean square error of an estimation procedure after a preliminarytest of signi�cance for three-way layout in a nova model II. Sankhya B 38, 164–178.

Thompson, J.R., 1968. Some shrinkage techniques for estimating the mean. J. Amer. Statist. Assoc. 63,113–123.

sometimes pool t estimation – via shrinkage

Documents