Statistical Papers 49, 37-58 (2008)

Statistical Papers © Springer-Verlag 2008

A modified estimator of population mean using power transformation

Housila P. Singh 1, Rajesh Tailor ~, Sarjinder Singh 2, Jong-Min Kim 3

School of Studies in Statistics, Vikram University, Ujjain - 456010, M. P., India

2 Department of Statistics, St. Cloud State University, St. Cloud, MN 56301 USA; (e-mail: sarjinder@yahoo.com)

3 Statistics, Division of Science and Mathematics, University of Minnesota - Morris, Morris, MN 56267, USA

Received: March 1, 2005; revised version: December 6, 2005

Summary

In this paper we have suggested two modified estimators of population mean using power transformation. It has been shown that the modified estimators are more efficient than the sample mean estimator, usual ratio estimator, Sisodia and Dwivedi ' s (1981) estimator and Upadhyaya and Singh's (1999) estimator at their optimum conditions. Empirical illustrations are also given for examining the merits of the proposed estimators. Following Kadilar and Cingi (2003) the work has been extended to stratified random sampling, and the same data set has been studied to examine the performance in stratified random sampling.

Keywords: Study variate, auxiliary variate, mean squared error,

efficiency.

38

I. Introduct ion

Consider a finite population u = ( u 1 , u 2 ..... UN) of size N. Let y and

x denote the study variate and the auxiliary variate taking values yi and xi respectively on the i th unit ui (i = 1,2 . . . . . U ). We assume

that (yi, xi) > 0, since survey variates are generally non-negative. Let N N - -2

Y'= N-1 Y~Yi, $2 = ( N - l ) -1Y~(Yi-Y) a n d Cy : Sy/F b e t h e populat ion i=1 i=1

mean, population variance and population coefficient of variation X

of the study variable y respectively. Further a s s u m e ~ = N -1 Y.xi, i=1

1N --4/ N x ~.2 2 S 2 : ( N _ I ) - y~(xi ,~)2 Cx sx/~andfl2(x): N - , : N Z ( x i - Y ) / { t~ l ( i - ) } i=t i=t "=

are the known population mean, population variance, population coefficient of variation and population coefficient of kurtosis of the auxiliary variable x respectively. Assume that a simple random sample of size n is drawn without replacement from population v . Let y and 2 be the sample means of y and x respectively. For

estimating population mean 7 the classical ratio estimator is defined as:

where the population mean x of the auxiliary variable is assumed to be known. If Y and cx are known, Sisodia and Dwivedi (1981)

suggested a transformed ratio type estimator for the population mean F as:

= y--('g+ Cx t (1.2)

In many practical situations the value of the auxiliary variate x may be available for each unit in the population, for instance, see Das and Tripathi (1980, 1981), Singh (2004), Stearns and Singh (2005) and Kadilar and Cigi (2005). Thus utilizing the information on T , cx and fl2(x) of the auxiliary variable, Upadhyaya and

Singh (1999) suggested the following ratio-type estimators for population mean 7 as:

39

and

_.(#& {u)+ Cx .~R(2) : Y/#"6'2 (x)+ Cx ] (1.3)

YR(3) = k-'( }TCx -+ ~ (x) / (1.4)

It is a well known result that the regression estimator is more efficient than the ratio (product) estimator except in the case where the regression line of the variable y on the variable x passes through the neighborhood of the origin, in which case the efficiencies of these estimators are almost equal. However, in many practical situations the regression line does not pass through the origin. Considering this fact Srivastava (1967) suggested a modified ratio-type estimator using power transformation is more efficient than the usual ratio estimator in some situations, and related work can be had from Singh (2003).

In the present investigation we have suggested the modification of Upadhyaya and Singh (1999) estimators by using the concept of power transformation earlier used by Srivastava (1967). It is shown that the proposed estimators are more efficient than the sample mean estimator y, usual ratio estimator JR(0), Sisodia and

Dwivedi (1981) estimator Ya(t), and Upadhyaya and Singh (1999)

estimators .~R(2) and YR(3). Numerical illustrations are given to

judge the merits of the suggested estimators over others.

2. The modified estimators

By applying power transformation on Upadhyaya and Singh (1999) estimators, the modified estimators are given by:

and

-.:~.~2(x)+Cx } a "~R(°:) = Y~ -~lq2 (x)+ Cx

_[ XCx +/5'2 (x)[ # y~(a) = y~ ~c+ + v= (x)J

(2.1)

(2.2)

where a and 6 are suitably chosen scalars such that the mean squared errors of Ya(~) and YR(a) are minimum. It may be noted

40

that the new estimators are generalizations of earlier estimators, namely (2.1) generalizes (1.3), and (2.2) generalizes (1.4).

To the first degree of approximation, the biases and mean squared errors of Ya(~) and Ye,(a) are, respectively, given by:

B(37R(o~))= (T/2)Aff0Cx 2 [(or + 1)0- 2K]

B~R(a })= (7/2)2o~0' Cx 2 [(6 + 1)o'-gK]

MSE(TVR(a) )= AY'g [c g + ~zoc g (~zO - 2K )]

and

where K = vCy/Cx, and p is the correlation coefficient between y and x.

(2.3)

(2.4)

(2.5)

MSE@R(a ))= .g~2 [Cy2 + 60'C 2 (60'-2K)] (2.6)

O = {.Xfig(x)}/{.~fi2(x)+ Cx} , tg'= {.~Cx}/{.~C x + fig(x)}, 2 = (N - n),/(nN),

The mean squared errors minimized for:

and

a=(K/O)=aop t (say)

at (2.5) and (2.6) are, respectively,

(2.7)

6 = (K/O')= 6op t (say) (2.8)

Thus the common minimum mean squared error of Ya(o,) and YR(8)

is given by:

min.MSE~R(a)) = min.MSE~R(6)) = A72C2 (1 - p 2 ) (2.9)

Substitution of (2.7) and (2.8) respectively in (2.3) and (2.4) yield the resulting biases of ~(~) and yp,(a) as:

and

B(YP@op,)) = ~(F/2)K(0- K)C 2 (2.10)

B(yp,(aopt)) = .¢(F/Z)K(O'-K)C 2 (2.11)

41

3. Efficiencies of modified estimators

It can easily be proved that the proposed modified estimator y~(~)

has lower mean squared error than the:

( i ) sample mean estimator y if:

2K 0 < ~ < (3.1)

0

( ii ) usual ratio estimator YR(0) if:

I o-Kl<lx-q that is if:

2 K - 1 either ( ) < c~ < ( - - - ~ ) when K > 1

2K - I O or ( - - - - -~) < ¢ < ( ) when K < 1

(3.2)

( iii ) Sisodia and Dwivedi's (1981) estimator YR0) if:

I=o-KI<IK- I that is if:

either (~-- - - )<c~<(2K-~) when K > ~ O 0

where

or (-~---~-) < a < (O) when K < g/

(3.3)

( iv ) Upadhyaya and Singh (1999) estimator YR(z) if:

I o-xl<lK-o I that is if:

2K - 0 either l < ~ z < ( - - - 7 ) when K > 0

2K - 0 or ( - - ) < a ' < l when K < 0

0

(3.4)

42

( v ) Upadhyaya and Singh (1999) estimator YR(3) if: I O-Kt<IK-O'I

that is if: _~ 2K - 0 '

either ( ) < ot < (------7-)

o r 2K - 0' _~ (---b--) < a < ( )

1 when K > 0' ]

f when K < 0'J

Further it can easily be proved that the proposed est imator Ya(a) has lower mean square error than the:

( i ) sample mean estimator .v if:

0 < J < (-~-)

( ii ) usual ratio estimator -FR(O) if: lae,-Kl<lK-lr

that is if:

1 2K -1 either (-~7) < ,5" < (---~;----) when K > 1

2 K - 1 1 or ( - - ) < a < ( - - ) when K <1

0'

( iii ) Sisodia and Dwived i (1981) estimator YR0) if:

[dO'-KI<IK-q/[

that is if:

t Y ,2K -g/. either ('--) < 6 < t - - - ~ - )

o' when K > ~,

2K -q / ~7, or ( - - 7 - - ) < a < ( ) when K < q /

( iv ) Upadhyaya and Singh (1999) estimator YR(2) if:

I o'-Kl<lx-ol that is if:

(3.5)

modif ied

(3.6)

(3.7)

(3.8)

43

2K - 0 t either (~-7) < 6 < (------7-) when K > 0

2K-t~ 0 K <O J or ( - - 7 - ) < 6 < (~-7) when

(3.9)

( v ) Upadhyaya and Singh (1999) estimator YR(3) if:

o'-KI < I x - o'1

that is if:

2K -0 ' either 1 < 6 < ( - - -~ - - ) when K > 0'

2K -8 ' or ( ~ ) < o ~ < I when K<O'

0 '

(3.10)

The proofs of the results (3.1) to (3.10) are simple, so omitted. It is to be noted that theMSE(YR(~)) reduces as the value of la-aopt I

decreases because the expression in (2.5) is a parabola in oz. Moreover the MSE(yR(~)) is minimized when I~-aoptl :0 by the

definition of aopt. Similar remarks apply to YR(5). The minimum

MSE in (2.9) is always less than the MSEs of Upadhyaya and Singh's estimators. This leads that the suggested estimators at their optimum conditions (i.e. a=~op~ and J=8opt) is more

efficient than Upadhyaya and Singh's estimators. For this we established the following theorems.

Theorem 3.1. At optimum c~ i.e. ~ =Crop t (8 i.e. 6= 8op t ) the

estimator YR(.) (and YR(~)) is better than Upadhyaya and Singh

(1999) estimators YR(2) (and YR(3)).

Proof. It follows immediately by the definition of O~op t (and 8op t ).

Theorem 3.2. At optimum a i.e. c~ = O(op t (• i.e. 8 = 8 o p t ) t h e

estimator Y'R(~) (and YR(~)) is more efficient than sample mean

estimator y , usual ratio estimator YR(0), and Sisodia and Dwivedi

(1981) estimator YR0)"

Proof. The proof of the theorem is simple, so omitted.

44

4. Empirical study using simple random sampling design

To examine the performance of the constructed estimators TR(,~)

and .7a(a) in comparison to the usual estimators, we have

considered two natural population data sets.

Population I: ]Source: Das 1988]

The population consists of 278 villages/towns/wards under Gajole Police Station of Malda district of West Bengal, India. The variates considered are:

and

y" the number of agricultural laborers in 1971

x" the number of agricultural laborers in 1961.

The values of required parameters are: T = 39.0680, x = 25.1110,

Cy =1.4451, C x =1.6198, p=0 .7213 , and f12(x)=38.8898. The percent

relative efficiencies (PREs) of different estimators with respect to sample mean estimator ~ have been computed and presented in

Tables 4.1, 4.2 and 4.3.

Table 4.1. PRE of YR(a) with respect to y for different values of o~.

O~ 0.00 0.25 0.50 a'op t = 0.64457

PRE 100.0 148.22 197.67 208.45 c~ 0.75 1.00 1.25 1.30 PRE 202.58 157.00 106.53 98.26

Table 4.2. PRE of YR(a) with respect to ~ for different values of ~ .

5 0.00 0.25 0.50 0.75 1.00 t.25

PRE 100.0 122.87 149.53 177.08 199.31 208.44 c~ 1.75 2.00 2.25 2.50 ~opt = 1.2588 1.50

PRE 178.90 151.49 124.64 101.46 208.45 200.47

Table 4.3. PREs of ~g(i), i = 0, 1, 2, 3 with respect to y .

Estimator "~ YR(0) YR(1) YR(2) YR(3) PRE 100.00 156.39 118.54 157.00 199.31

Table 4.4. Ranges of o~ and c~ for YR(a) and -VR(a) to be more

efficient than different estimators of population Estimator Range of o~

mean. Range of 6

y (0.0000, 1.2891) (0.0000, 2.5176) JR(O) (0.2865, 0.9983) (0.5614, 1.9562)

J R ( I ) (0.3470, 0.8825) (0.6800, 1.8377)

YR(2) (0.2848, 1.000) (0.5581, 1.9593)

YR(3) (0.5104, 0.7743) (1.0000, 1.5176)

45

From Tables 4.1, 4.2, 4.3 and 4.4, we observe that there is considerable gain in efficiency by using ~(~) and Ye,(a) over j ,

JR(0), Je,0), Ye,(2) and Ja(3). There is also enough scope of

choosing a and 8 to obtain better estimators from YR(~) and YR(a)

respectively. When cc~(0.5104,0.7743) and ae(1.0000, 1.5176), the

proposed modified estimators YR(~) and Ye,(a) are always more

efficient than y and JR(0, i = 0,1, 2, 3, respectively.

Population II: [Source: Cochran (1977, pp. 325)]

The variates are defined as follows:

and

y • Number of persons per block

x" Number of rooms per block.

The values of different required parameters are: ~-=101.10,

~=58.80, Cy=0.1445, Cx=0.1281 , /7=0.6500 and f12(x)=2.2387. The

percent relative efficiencies (PREs) of different estimators with respect to sample mean estimator y have been computed and listed in Tables 4.5, 4.6, 4.7 and 4.8.

Table 4.5. PRE of YR(a) with respect to sample mean

estimator ~ for different values of a ' .

a' 0.00 0.25 0.50 O'op t = 0.73393

PRE 100.00 131.37 161.18 173.16

c~ 0.75 1.00 1.25 1.50 PRE 173.10 157.97 127.16 96.36

46

Table 4.6. PRE of FR(a) with respect to sample mean

estimator y for different values of 8.

6 0.00 0.25 0.50 -PRE 100.00 123.90 148.69

6 6op t = 0.95112 1.00 1.25

PRE 173.16 172.87 161.50

0.75 167.67

1.50

139.24

Table 4.7. Percent relative efficiencies of sample mean estimator T and JR(i), i = 0, 1, 2, 3 with respect to sample mean y.

Estimator Y ½(o) Y~(1) I 'R(2 ) 'R(3) t PRE 100.00 158.23 158.45 157.97 172.87

more efficient Table 4.8. Range of c~ and 6 for fiR(a) and .~e,(8) to be than different estimators of population mean.

Estimator Range of ~ Range of 6 fi (0.0000, 1.4679) (0.0000, 1.9022)

YR(0) (0.4684, 1.00 I0) (0.6050, 1.2972)

-~R0) (0.4691, 0.9978) (0.6079, 1.2944)

YR(2) (0.4679, 1.0000) (0.6063, 1.2958) Ya(3) (0.6962, 0.7717) (0.9022, 1.0000)

Tables 4.4, 4.5 and 4.6 show that the proposed estimators YR(~z) and YR(~) are more efficient than the estimators y and YR(i), i= 0,1, 2,3 at their optimum conditions. Even when c~ and d depart

from their optimum values (aop t and 8opt ), the proposed estimators

YR(~) and Ya(8) beat the estimators under reference. When

a ~ (0.6962, 0.7717) and 6 e (0.9022, 1.0000), the estimators YR(~) and

YR(6) are always better than y and YR(i), i= 0,1,2,3, respectively.

The range of 8 is very small in which YR(8) is better than y and

:TR(i), i = 0 ,1 ,2 ,3 , as the estimator YR(3) happens to be almost

equally efficient to the estimator YR(8) at c~ = ~opt.

The next section has been used to extend these estimators in stratified random sampling, because the stratified sampling has more practical utility. The next section also answers a recent question raised by Kadilar and Cingi (2003) about the performance of such estimators in stratified random sampling.

47

5. Stratified random sampling

To save the space we strictly follow the notations of Kadilar and k k

Cingi (2003, 2005). Let Yst = EcohYh, and ~st = Z~h~h, where k is h=l h=l

the number of strata, con =Nh/N is the stratum weight, N h be the k

number of units in the h th stratum such that N = Z N~ be the total h=l

population size, Yh is the sample mean of variate of interest in

stratum h and Yh is the sample mean of the auxiliary variate x in

stratum h. The usual combined ratio estimator in stratified random sampling is given by:

\ Xst /

The exact variance of Yst, and to the first degree of approximation

the mean squared error of YRc are, respectively, given by:

and

VOSst)= ~ co 2 S 2 h=l hYh yh

h=l

(5.2)

(5.3)

where yh ={1-(nh/Nh)}/nh, R=F/Y is the population ratio, n h is the

number of units in h ~h stratum, s : is the population variance of the yh

variate of interest y in the h th stratum, s~h is the population

variance of the auxiliary variable x in the h th stratum, and Syxh is

the population covariance between auxiliary variate x and variate of interest y in the h th stratum. In stratified random sampling,

Kadilar and Cingi (2003) suggested the following estimators for as:

YstSD = YstI XsD1 (5.4) \ XSD /

ystsK = y~t( xsx l (5.5) \ xsr~ /

48

and

#stus, = #st( Xus, I (5.6) \ xusI J

Ystus2 = Y~t( xUs21 (5.7) \ xus2 j

where k _ k _ k

z..,,(~,, ,~.,,(x))..,~ z..,,(~-,,+c.,,,), ~ : zo.,,(7,, +Cx~),x~ = + = h=l h=l h=l

k _ k _ ~ , : z ~ ( ~ . ~ ( ~ ) + C x ~ ) . ~ . : z ~ ( ~ h c ~ + ~(~)),

h=l h=l

k k XSK = ~.O)h('~ h + fl2h(X)), XUS1 : ~,gOh(.~hflZh(X)+Cxh), Cxh = S x h / Y h ,

h=l h=l

1 1 h=l

and Xh = Nh 1 N~ h Xhj have their usual meanings. j = l

To the first degree of approximation the mean squared errors of the estimators YstSD, YstSK, Y'~USl and Y~tusz are respectively given by:

k 2 (5.s)

MSE(~stSK) = ~ O)27h(S2h-2RSKSyxh h=l

(5.9)

R2 ~2 (x~s 2 ~ (5.10) h=l

and 2 2 2

MSE(YstUS2)= h=l ~ co2hYh[" (S2yh -2Rus2CxhSyxh+Rus2CxhSxh) (5.11)

where

k k Z O)hYh 2 0)hY'h

RUS1 = h=l and R U S 2 = h=l k k _ v. 0.,, [rh ~,h (,0+ cx. ] x 0.,, [~ hcx~ + ,~.,, (x)]

h=l h=l

49

Through an empirical study Kadilar and Cingi (2003) have shown that the traditional combined ratio estimator YRc is more efficient

than the ratio estimators Y,tsD, YstSX, Ysttzsl and .~stUS2; and remarked that, in the forthcoming studies, new ratio-type estimators should be proposed not only in simple random sampling but also in stratified random sampling. This led authors of this paper to suggest some modified ratio type estimators in stratified sampling.

6. Modified estimators in stratified random sampling

We suggest the modified estimators for the population mean P- as:

and

f g "~ C('SI

. . . . \ xgsl ) (6.1)

xus2 ) (6.2)

where ast and 6~, are suitably chosen scalars. For

(ast,,s~,)= (0,0), 0,1); the estimators YR(~s,) and YR(a,.,) are respectively

reduced to Y~t and (Y~tusl, Y~tc~s2).

To the first degree of approximation, the mean squared errors of YR(~s,) and YR(as,) are respectively given by:

hZh~ yh ~stRUSlflZh(X)Sxh - 2°~stRUSlfl2h(X)Syxh) (6.3) h = l

and k 2 - 2 2 2 2 2 MSE~R(dst))= Y~ COhYh(Svh+dstRus2CxhSxh-26stRUS2CxhSyxh ) (6.4)

h= l " - /

From (5.2), (5.3), (5.8)-(5.11) and (6.3) we note that the estimator YR(~,,) is more efficient than:

( i ) .Ss t if: 2 A

0 < ,Zst < (--77---) BI~us 1

(6.5)

~,.~

o

I A I

"4

I A A

~.

J

A

A

~

A

A

: *x

I

%

-

,.

.,

v

~-'

J

A

V

~

~.-,

o

I A I

I A

A A

A

A I

J

v

I A I

oo

ox

b~

* A C~

A *

I A

~ ,.

. j

>

A

p>

v

I-O

~.=.

# I A i

f~

A A

~.-~.

~.,~°

O _.

ixJ

I::r

:m

~...

©

II

r

I rr

"m0

i A A ÷ cm

~zr

I I I A

o

p..~.

A A A

A I V

~°

I A I

A

A

-t-

I I I b-~

i J

A

A

A

,J

A

A A I

J

v

I A I

I A A

J

A

A A I V

I A I

~.~

° ~

o

L~

L~

53

where

and

either l < f i s t | - 1 when . ,A___ >1 < ,

B R US2

o r / A* 1 -1 <6st <1 when - - - <1 \ B RUS 2 B * R u s 2

, k 2 B* k 2 2 2 A = ~ COhYhCxhSyxh , = Y~ COhYhCxhSxh ,

h=l h=l

I ( 'Rusl ) BR - 2A) 1

(6.16)

The proofs of the results in (6.5) to (6.16) are simple, so omitted. The minimization of the MSEs (6.3) and (6.4) with respect to est

and fist, respectively, we get the optimum values of c~t and 8st as:

c~!t ) _ A (6.17) BRus1

and

8!~')= ,A (6.18) B RUS 2

Thus the minimum MSE of YR(~,,) and

given by:

min.MSE(yR(ast))=Ih~=lOa2yhS2h-A-~--~l

and

min.MS E (.~ R(,~st ) ) =Lh=l h ~/h yh B* J "

YR(4~,) are respectively

(6.19)

(6.20)

From (5.2), (5.10) and (6.19) we have:

and

A 2 V(Yst ) -min 'MSE(27R(a ' t ) )= B >0

(A- Rus1B) 2 > 0 MSE(fistUS1 ) - minMSE('vR(as')) - B

(6.21)

(6.22)

54

Thus from (6.21) and (6.22) it follows that the estimator YR(~s,) is

more efficient than the estimator Ys,sel. Further from (5.2), (5.11)

and (6.20) we have:

and

A*2 V(Yst ) - m i n ' M S E ( y R ( a s , ) ) = - 7 - -> 0 (6.23)

B

(A* ,)2 MSE(~stUS1 )_ min .MSE( .~R(¢ ' ) ) = - RUS 2 , B > 0

B (6.24)

We note from (6.23) and (6.24) that the estimator YR(as,) is better

than usual unbiased estimator Y~, and the estimator Ys,SV2. In

practice, if the values of . ! t ) and @) are not known, it is

advisable to use their consistent estimators as:

and

d,!~') - ~- "~ (6.25) BRus1

@ ) - ~,= (6.26) B RUS 2

where: k k ^ k

= , = o92 o2 [x~s 2 ]4 Z °)2)'hfl2h(X)Syxh B X hYhP2h~ ) xh , A = Z O)2yhCxhSyxh, h=l h=l h=l

nh ~ , ~' 2 2 2 2 (~h-O-~(xhj ~h) 2 Yh='qlzYhj, = Y.(.OhYhCxhSxh , Sxh = - , h=l j=l j= l

RUS1 2 1 h Y h h{-Xh,82h(x)+Cxh}' Syxh ( n h - 1 ) - l n h = = E (Yh j -Yh) (Xh j - -Yh) , j= t

-Xh = nh I Z and ~ s z = Zc°hYh h{-XhCxh + fl2h(X)} • j= l = =

Thus we get the resulting estimators for the population mean F as:

and

- ~ _ - ( x u s l ) a~;)

V stl ~, xUS 2 J

(6.27)

(6.28)

55

To the first degree of approximation it can be shown that:

and

MSE ~R(~s ,)) min.MSE~R(a~s,))

MSE(.~R(a~,)] = min.MSE(~R(as,)) •

(6.29)

(6.30)

where min.MSE~R(a,.,) ) and min.MSE~R(as,)) are respectively given

by (6.19) and (6.20).

7. Empirical study using stratified random sampling

In order to see the performance of the suggested estimators over other estimators, we have chosen the same data set as considered by Kadilar and Cingi (2003) and is related to biometrical science. The percent relative efficiencies (PREs) of different estimators with respect to Ys, have been computed and presented in Tables

7.1, 7.2, 7.3 and 7.4.

Table 7.1. PRE of YR(c~st) with respect to Yst for different values of ast. test 0.0 0.25 0.50 0.75 g!t ) 0.82579

PRE 100.00 144.24 206.62 245.03 248.09 ~Zst 1.00 1.25 1.50 1.60 1.6515

PRE 232.75 178.38 124.85 107.79 100.00

Table 7.2. PRE of ~R(ssl) with respect to Yst for different values of 8st.

6st 0.00 0.25 0.50 0.75 1.00 PRE 100.00 141.57 202.91 278.09 326.37 fist ~;!~) = 1.0411 1.25 1.50 1.75 2.00 2.0822

PRE 360.09 300.00 227.20 159.41 117.78 100.00

Tables 7.1 and 7.2 exhibit that the estimators YR(as,) and YR(4,) are

better than the conventional unbiased estimator fist even when the

scalars o~st and as, depart much from their corresponding optimum

values c~!7) and @). Thus there is enough scope of choosing

56

scalars ~st and 3",t in YR(~,~) and YR(8,,) to obtain better estimators.

It is further observed from the Table 7.3 that the estimators -YR(a~.,)

and YR(~st) (or the estimators YR(ast) and YR(g~) based on estimated

optimum values) give largest gain in efficiency at their optimum -(o ;(o) ). conditions (i.e. the optimum estimators YRI~,t) and R(~.t)

Table 7.3. PREs of .~st, ~Rc, YstSD, YstSK, .~stUS1, YstUS2, f(~lCtst )

(or ~(R0/&s,)], ~(R0/fis,)(or ~(R0~s,)] with respect to fist.

Estimator Yst -~ RC ~stSD YstSK

PRE 100.00 312.21 312.00 312.02

Estimator YstUS1 YstUS2 y(~tCtst ) ~(ROIsst )

PRE 232.75 326.37 248.09 360.09

We have further computed the ranges of c~t and 6~t for Yst(~s,)

and f~t(4~,) to be more efficient than different estimators of

population mean Y and compiled in Table 7.4.

Table 7.4. Ranges of ~st and •st for Yst(ast ) and Yst(J~,) to be more

efficient than various estimators of the population mean. Estimator Range of ~st Range of 6st

Yst (0.0000, 1.6516) (0.0000, 2.0822) YRC (-54.0127, 55.6643) (-0.6522, 2.7344)

YstSD (-54.0056, 55.6571) (-0.6518, 2.7340) ~stSK (-53.9484, 55.6000) (-0.6490, 2.7312) "TstUSl (0.6516, 1.0000) (0.6004, 1.4818) YstUS2 (0.0000, 1.6515) (1.0000, 1.0822)

Table 7.4 gives common range of c~s, as (0.6516, 1.0000) for

YR(~st) to be better than the estimators Yst, .YRC, Y~tSD, Y~tS*:,

.~stUSi, i=1, 2 while the common range of ~st is (1.0000, 1.0822)

for YR(Sst) to be more efficient than the rest of the estimators.

57

Conclusions

We conclude that the modified estimators YR(a) and Ya(g), and

their extension in stratified sampling are worth using not only at their optimum conditions, for in a quite wide range of scalars around the optimum conditions. Thus this study answers a valuable question recently raised by Kadilar and Cingi (2003) about the doubtfulness of the validity of the theory of ratio type estimators in stratified random sampling and simple random sampling.

Acknowledgements

The authors are thankful to the Editor Professor G6tz Trenkler and a learned referee for their valuable comments to bring the original manuscript in the present form.

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