A modified estimator of population mean using power transformation

Transcription

1 Statistical Papers 49, (2008) Statistical Papers Springer-Verlag 2008 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 , M. P., ndia 2 Department of Statistics, St. Cloud State University, St. Cloud, MN US; ( sarjinder@yahoo.com) 3 Statistics, Division of Science Mathematics, University of Minnesota - Morris, Morris, MN 56267, US Received: March 1, 2005; revised version: December 6, 2005 Summary n this paper we have suggested two modified estimators of population mean using power transformation. t has been shown that the modified estimators are more efficient than the sample mean estimator, usual ratio estimator, Sisodia Dwivedi's (1981) estimator Upadhyaya Singh's (1999) estimator at their optimum conditions. Empirical illustrations are also given for examining the merits of the proposed estimators. Following Kadilar Cingi (2003) the work has been extended to stratified rom sampling, the same data set has been studied to examine the performance in stratified rom sampling. Keywords: Study variate, auxiliary variate, mean squared error, efficiency.

2 38. ntroduction Consider a finite population u =(u1,u 2... UN) of size N. Let y x denote the study variate the auxiliary variate taking values yi 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) Cy : Sy/F be the population i=1 i=1 mean, population variance population coefficient of variation X of the study variable y respectively. Further assume ~=N -1 Y.xi, i=1 1N --4/ N x ~.2 2 S2:(N_) - y~(xi,~)2 Cx sx/~fl2(x): N -, : NZ(xi-Y) /{t~l ( i- ) } i=t i=t "= are the known population mean, population variance, population coefficient of variation population coefficient of kurtosis of the auxiliary variable x respectively. ssume that a simple rom sample of size n is drawn without replacement from population v. Let y 2 be the sample means of y 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. f Y cx are known, Sisodia Dwivedi (1981) suggested a transformed ratio type estimator for the population mean F as: = y--('g+ Cx t (1.2) n many practical situations the value of the auxiliary variate x may be available for each unit in the population, for instance, see Das Tripathi (1980, 1981), Singh (2004), Stearns Singh (2005) Kadilar Cigi (2005). Thus utilizing the information on T, cx fl2(x) of the auxiliary variable, Upadhyaya Singh (1999) suggested the following ratio-type estimators for population mean 7 as:

3 39 _.(#& {u)+ Cx.~R(2): Y/#"6'2 (x)+ Cx ] (1.3) YR(3) = k-'( }TCx -+ ~ (x) / (1.4) t 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, related work can be had from Singh (2003). n the present investigation we have suggested the modification of Upadhyaya Singh (1999) estimators by using the concept of power transformation earlier used by Srivastava (1967). t is shown that the proposed estimators are more efficient than the sample mean estimator y, usual ratio estimator JR(0), Sisodia Dwivedi (1981) estimator Ya(t), Upadhyaya Singh (1999) estimators.~r(2) 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 Singh (1999) estimators, the modified estimators are given by: -.:~.~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 6 are suitably chosen scalars such that the mean squared errors of Ya(~) YR(a) are minimum. t may be noted

4 40 that the new estimators are generalizations of earlier estimators, namely (2.1) generalizes (1.3), (2.2) generalizes (1.4). To the first degree of approximation, the biases mean squared errors of Ya(~) Ye,(a) are, respectively, given by: B(37R(o~))= (T/2)ff0Cx 2 [(or + 1)0-2K] B~R(a })= (7/2)2o~0' Cx 2 [(6 + 1)o'-gK] (2.3) (2.4) MSE(TVR(a) )= Y'g [c g + ~zoc g (~zo - 2K )] (2.5) MSE@R(a ))=.g~2 [Cy2 + 60'C 2 (60'-2K)] (2.6) where K = vcy/cx, p is the correlation coefficient between y x. O = {.Xfig(x)}/{.~fi2(x)+ Cx}, tg'= {.~Cx}/{.~C x + fig(x)}, 2 = (N - n),/(nn), The mean squared errors minimized for: a=(k/o)=aop t (say) at (2.5) (2.6) are, respectively, (2.7) 6 = (K/O')= 6op t (say) (2.8) Thus the common minimum mean squared error of Ya(o,) YR(8) is given by: min.mse~r(a)) = min.mse~r(6)) = 72C2 (1 - p 2 ) (2.9) Substitution of (2.7) (2.8) respectively in (2.3) (2.4) yield the resulting biases of ~(~) yp,(a) as: B(YP@op,)) = ~(F/2)K(0- K)C 2 (2.10) B(yp,(aopt)) =. (F/Z)K(O'-K)C 2 (2.11)

5 41 3. Efficiencies of modified estimators t 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: o-kl<lx-q that is if: 2K-1 either ( ) < c~ < (---~) when K > 1 or (-----~) 2K - < < ( O ) when K < 1 (3.2) ( iii ) Sisodia Dwivedi's (1981) estimator YR0) if: =o-k<k- that is if: either (~----)<c~<(2k-~) when K>~ O 0 (3.3) or (-~---~-) < a < (O) when K < g/ where ( iv ) Upadhyaya Singh (1999) estimator YR(z) if: o-xl<lk-o that is if: 2K -0 either l<~z<(---7) when K>0 2K -0 or (--)<a'<l when K<0 0 (3.4)

6 42 ( v ) Upadhyaya Singh (1999) estimator YR(3) if: O-Kt<K-O' that is if: _~ 2K -0' either ( ) < ot < ( ) or (---b--) 2K - 0' < a < (_~) 1 when K > 0' ] when f K < 0'J (3.5) Further it can easily be proved that the proposed estimator Ya(a) has lower mean square error than the: modified ( i ) sample mean estimator.v if: 0 < J < (-~-) (3.6) ( ii ) usual ratio estimator -FR(O) if: lae,-kl<lk-lr that is if: 1 2K -1 either (-~7) <,5" < (---~;----) when K > 1 2K-1 1 or (--) < a<(--) when K <1 0' (3.7) ( iii ) Sisodia Dwivedi (1981) estimator YR0) if: that is if: [do'-k<k-q/[ ty,2k -g/. either ('--) < 6 < t---~-) o' when K > ~, 2K -q/ ~7, or (--7--)<a<( ) when K<q/ (3.8) ( iv ) Upadhyaya Singh (1999) estimator YR(2) if: o'-kl<lx-ol that is if:

7 43 either (~-7) < 6 < ( ) 2K -0 when K > 0 t 2K-t~ 0 K <O J or (--7-) < 6 < (~-7) when (3.9) ( v ) Upadhyaya Singh (1999) estimator YR(3) if: o'-k < x - o'1 that is if: 2K -0' either 1 < 6 < (---~--) when K > 0' 2K -8' or (~)<o~< when K<O' 0 ' (3.10) The proofs of the results (3.1) to (3.10) are simple, so omitted. t is to be noted that themse(yr(~)) reduces as the value of la-aopt decreases because the expression in (2.5) is a parabola in oz. Moreover the MSE(yR(~)) is minimized when ~-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 Singh's estimators. This leads that the suggested estimators at their optimum conditions (i.e. a=~op~ J=8opt) is more efficient than Upadhyaya Singh's estimators. established the following theorems. Theorem 3.1. t optimum c~ i.e. ~ =Crop t (8 For this we i.e. 6= 8op t ) the estimator YR(.) ( YR(~)) is better than Upadhyaya Singh (1999) estimators YR(2) ( YR(3)). Proof. t follows immediately by the definition of O~op t ( 8op t ). Theorem 3.2. t optimum a i.e. c~ = O(op t ( i.e. 8=8opt)the estimator Y'R(~) ( YR(~)) is more efficient than sample mean estimator y, usual ratio estimator YR(0), Sisodia Dwivedi (1981) estimator YR0)" Proof. The proof of the theorem is simple, so omitted.

8 44 4. Empirical study using simple rom sampling design To examine the performance of the constructed estimators TR(,~).7a(a) in comparison to the usual estimators, we have considered two natural population data sets. Population : ]Source: Das 1988] The population consists of 278 villages/towns/wards under Gajole Police Station of Malda district of West Bengal, ndia. The variates considered are: y" the number of agricultural laborers in 1971 x" the number of agricultural laborers in The values of required parameters are: T = , x = , Cy =1.4451, C x =1.6198, p=0.7213, f12(x)= The percent relative efficiencies (PREs) of different estimators with respect to sample mean estimator ~ have been computed presented in Tables 4.1, Table 4.1. PRE of YR(a) with respect to y for different values of o~. O~ a'op t = PRE c~ PRE Table 4.2. PRE of YR(a) with respect to ~ for different values of ~ t.25 PRE c~ ~opt = PRE 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

9 45 Table 4.4. Ranges of o~ c~ for YR(a) -VR(a) to be more efficient than different estimators of population mean. Estimator Range of o~ Range of 6 y (0.0000, ) (0.0000, ) JR(O) (0.2865, ) (0.5614, ) JR() (0.3470, ) (0.6800, ) YR(2) (0.2848, 1.000) (0.5581, ) YR(3) (0.5104, ) (1.0000, ) From Tables 4.1, 4.2, , we observe that there is considerable gain in efficiency by using ~(~) Ye,(a) over j, JR(0), Je,0), Ye,(2) Ja(3). There is also enough scope of choosing a 8 to obtain better estimators from YR(~) YR(a) respectively. When cc~(0.5104,0.7743) ae(1.0000, ), the proposed modified estimators YR(~) Ye,(a) are always more efficient than y JR(0, i = 0,1, 2, 3, respectively. Population : [Source: Cochran (1977, pp. 325)] The variates are defined as follows: 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= f12(x)= The percent relative efficiencies (PREs) of different estimators with respect to sample mean estimator y have been computed listed in Tables 4.5, 4.6, Table 4.5. PRE of YR(a) with respect to sample mean estimator ~ for different values of a'. a' O'op t = PRE c~ PRE

10 46 Table 4.6. PRE of FR(a) with respect to sample mean estimator y for different values of PRE op t = PRE Table 4.7. Percent relative efficiencies of sample mean estimator T JR(i), i = 0, 1, 2, 3 with respect to sample mean y. Estimator Y ½(o) Y~(1) 'R(2 ) 'R(3) t PRE Table 4.8. Range of c~ 6 for fir(a).~e,(8) to be more efficient than different estimators of population mean. Estimator Range of ~ Range of 6 fi (0.0000, ) (0.0000, ) YR(0) (0.4684, ) (0.6050, ) -~R0) (0.4691, ) (0.6079, ) YR(2) (0.4679, ) (0.6063, ) Ya(3) (0.6962, ) (0.9022, ) Tables 4.4, show that the proposed estimators YR(~z) YR(~) are more efficient than the estimators y YR(i), i= 0,1, 2,3 at their optimum conditions. Even when c~ d depart from their optimum values (aop t 8opt ), the proposed estimators YR(~) Ya(8) beat the estimators under reference. When a ~ (0.6962, ) 6 e (0.9022, ), the estimators YR(~) YR(6) are always better than y YR(i), i= 0,1,2,3, respectively. The range of 8 is very small in which YR(8) is better than y :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 rom sampling, because the stratified sampling has more practical utility. The next section also answers a recent question raised by Kadilar Cingi (2003) about the performance of such estimators in stratified rom sampling.

11 47 5. Stratified rom sampling To save the space we strictly follow the notations of Kadilar k Cingi (2003, 2005). Let Yst = EcohYh, ~st = Z~h~h, where k is 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 population size, Yh is the sample mean of variate of interest in stratum h Yh is the sample mean of the auxiliary variate x in stratum h. The usual combined ratio estimator in stratified rom sampling is given by: k \ Xst / The exact variance of Yst, to the first degree of approximation the mean squared error of YRc are, respectively, given by: VOSst)= ~ co 2 S 2 hyh yh (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: yh is the population variance of the 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, Syxh is the population covariance between auxiliary variate x variate of interest y in the h th stratum. n stratified rom sampling, Kadilar Cingi (2003) suggested the following estimators for as: YstSD = Yst XsD1 (5.4) \ XSD / ystsk = y~t( xsx l (5.5) \ xsr~ /

12 48 where #stus, = #st( Xus, (5.6) \ xus J Ystus2 = Y~t( xus21 (5.7) \ xus2 j k _ k _ k ~: zo.,,(7,, +Cx~),x~ = z..,,(~,, +,~.,,(x))..,~ = z..,,(~-,,+c.,,,), k _ k _ ~, : z~(~.~(~)+cx~). ~. : z~(~hc~ + ~(~)), k k XSK = ~.O)h('~ h + fl2h(x)), XUS1 : ~,goh(.~hflzh(x)+cxh), Cxh = Sxh/Yh, 1 1 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 Y~tusz are respectively given by: k 2 (5.s) MSE(~stSK) = ~ O)27h(S2h-2RSKSyxh (5.9) where MSE(YstUS2)= ~ co2hyh[" R2 ~2 (x~s 2 ~ (5.10) (S2yh -2Rus2CxhSyxh+Rus2CxhSxh) (5.11) k Z O)hYh k 2 0)hY'h RUS1 = RUS2 = k k _ v. 0.,, [rh ~,h (,0+ cx. ] x 0.,, [~ hcx~ +,~.,, (x)]

13 49 Through an empirical study Kadilar Cingi (2003) have shown that the traditional combined ratio estimator YRc is more efficient than the ratio estimators Y,tsD, YstSX, Ysttzsl.~stUS2; remarked that, in the forthcoming studies, new ratio-type estimators should be proposed not only in simple rom sampling but also in stratified rom sampling. This led authors of this paper to suggest some modified ratio type estimators in stratified sampling. 6. Modified estimators in stratified rom sampling We suggest the modified estimators for the population mean P- as: f g "~ C('S... \ xgsl ) xus2 ) (6.1) (6.2) where ast 6~, are suitably chosen scalars. For (ast,,s~,)= (0,0), 0,1); the estimators YR(~s,) YR(a,.,) are respectively reduced to Y~t (Y~tusl, Y~tc~s2). To the first degree of approximation, the mean squared errors of YR(~s,) YR(as,) are respectively given by: hzh~ yh ~struslflzh(x)sxh - 2 ~struslfl2h(x)syxh) (6.3) - k MSE~R(dst))= Y~ COhYh(Svh+dstRus2CxhSxh-26stRUS2CxhSyxh " - / ) (6.4) From (5.2), (5.3), (5.8)-(5.11) (6.3) we note that the estimator YR(~,,) is more efficient than: (i).sst if: 2 0 <,Zst < ( ) B~us 1 (6.5)

14 b~ v J "4 ~,.~o ~.-,o ~ : *x % -,.., v ~. J ~-' J V ~ oo ox

15 o p..~. V f~ ~.-~. ~.,~ ~zr ~.=. O r * C~ # _. ixj i * i ~,.. j ::r rr "m0 > p> :m v ~... cm -O

16 L~ L~ J V ~.~ ~o ~ -t- b-~ i J,J v J

17 53 where either l<fist <, -1 when., >1 B R US2 or / -1 <6st <1 when -- * - 1 <1 \ B RUS 2 B*Rus 2, k 2 B* k = ~ COhYhCxhSyxh, = Y~ COhYhCxhSxh, ( 'Rusl ) BR - 2) 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) (6.4) with respect to est fist, respectively, we get the optimum values of c~t 8st as: c~!t ) _ (6.17) BRus1 8!~')=, (6.18) B RUS 2 Thus the minimum MSE of YR(~,,) given by: min.mse(yr(ast))=h~=loa2yhs2h--~--~l YR(4~,) are respectively (6.19) min.ms E (.~ R(,~st ) ) =L h ~/h yh B* J " (6.20) From (5.2), (5.10) (6.19) we have: 2 V(Yst)-min'MSE(27R(a't))= B >0 (- Rus1B) 2 > 0 MSE(fistUS1 )- minmse('vr(as')) - B (6.21) (6.22)

18 - ~ 54 Thus from (6.21) (6.22) it follows that the estimator YR(~s,) is more efficient than the estimator Ys,sel. Further from (5.2), (5.11) (6.20) we have: *2 V(Yst )- min'mse(yr(as,)) = -7- -> 0 (6.23) B MSE(~stUS1 )_ min.mse(.~r( ')) = (*,)2 - RUS 2, B > 0 B (6.24) We note from (6.23) (6.24) that the estimator YR(as,) is better than usual unbiased estimator Y~, the estimator Ys,SV2. n practice, if the values of.!t are not known, it is advisable to use their consistent estimators as: where: d,!~') - ~- "~ (6.25) ~,= (6.26) B RUS 2 k k ^ k ]4 = Z )2)'hfl2h(X)Syxh, B = X o92 hyhp2h~ o2 [x~s ) xh, 2 = Z O)2yhCxhSyxh, nh = Y.(.OhYhCxhSxh, Sxh = -, j=l j=l ~, ~' (~h-o-~(xhj ~h) 2 Yh='qlzYhj, RUS1 = 21hYh h{-xh,82h(x)+cxh}' Syxh = (nh-1) -lnh E (Yhj-Yh)(Xhj--Yh), j=t -Xh = nh Z ~sz = Zc hyh h{-xhcxh + fl2h(x)} j=l = = Thus we get the resulting estimators for the population mean F as: _- (xusl) a~;) (6.27) V stl ~, xus 2 J (6.28)

19 55 To the first degree of approximation it can be shown that: MSE ~R(~s,)) min.mse~r(a~s,)) MSE(.~R(a~,)] = min.mse(~r(as,)) (6.29) (6.30) where min.mse~r(a,.,) ) min.mse~r(as,)) are respectively given by (6.19) (6.20). 7. Empirical study using stratified rom sampling n order to see the performance of the suggested estimators over other estimators, we have chosen the same data set as considered by Kadilar Cingi (2003) is related to biometrical science. The percent relative efficiencies (PREs) of different estimators with respect to Ys, have been computed presented in Tables 7.1, 7.2, Table 7.1. PRE of YR(c~st) with respect to Yst for different values of ast. test g!t ) PRE ~Zst PRE Table 7.2. PRE of ~R(ssl) with respect to Yst for different values of 8st. 6st PRE fist ~;!~) = PRE Tables exhibit that the estimators YR(as,) YR(4,) are better than the conventional unbiased estimator fist even when the scalars o~st as, depart much from their corresponding optimum values Thus there is enough scope of choosing

20 56 scalars ~st 3",t in YR(~,~) YR(8,,) to obtain better estimators. t is further observed from the Table 7.3 that the estimators -YR(a~.,) YR(~st) (or the estimators YR(ast) YR(g~) based on estimated optimum values) give largest gain in efficiency at their optimum -(o ;(o) ). conditions (i.e. the optimum estimators YR~,t) 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 Estimator YstUS1 YstUS2 y(~tctst ) ~(ROsst ) PRE We have further computed the ranges of c~t 6~t for Yst(~s,) f~t(4~,) to be more efficient than different estimators of population mean Y compiled in Table 7.4. Table 7.4. Ranges of ~st st for Yst(ast ) Yst(J~,) to be more efficient than various estimators of the population mean. Estimator Range of ~st Range of 6st Yst (0.0000, ) (0.0000, ) YRC ( , ) ( , ) YstSD ( , ) ( , ) ~stsk ( , ) ( , ) "TstUSl (0.6516, ) (0.6004, ) YstUS2 (0.0000, ) (1.0000, ) Table 7.4 gives common range of c~s, as (0.6516, ) 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, ) for YR(Sst) to be more efficient than the rest of the estimators.

21 57 Conclusions We conclude that the modified estimators YR(a) Ya(g), 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 Cingi (2003) about the doubtfulness of the validity of the theory of ratio type estimators in stratified rom sampling simple rom sampling. cknowledgements The authors are thankful to the Editor Professor G6tz Trenkler a learned referee for their valuable comments to bring the original manuscript in the present form. References Cochran, W.G. (1977). Sampling Techniques. John Wiley Sons. nc. London. Das,.K. (1988). Contribution to the theory of sampling strategies based on auxiliary information. Ph.D. thesis submitted to Bidhan Chra Krishi Vishwavidyalaya, Mohanpur, Nadia, West Bengal, ndia. Das,.K. Tripathi, T.P. (1980). Sampling strategies for population mean when the coefficient of variation of an auxiliary character is known. Sankhya, 42, C, Das,.K. Tripathi, T.P. (1981). class of sampling strategies for population mean using information on mean variance of an auxiliary character. Proc. of the ndian Statistical nstitute Golden Jubilee nternational Conference on Statistics." pplications New directions, Calcutta, December 1981, Kadilar, C. Cingi, H. (2003). Ratio estimators in stratified rom sampling. Biom. J., 45, 2,

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