ON THE FAMILY OF ESTIMATORS OF POPULATION MEAN IN STRATIFIED RANDOM SAMPLING

Transcription

1 Pak. J. Stat. 010 Vol. 6(), ON THE FAMIY OF ESTIMATORS OF POPUATION MEAN IN STRATIFIED RANDOM SAMPING Nursel Koyuncu and Cem Kadlar Hacettepe Unversty, Department of Statcs, Beytepe, Ankara, Turkey Emal: 1 nkoyuncu@acettepe.edu.tr and kadlar@acettepe.edu.tr ABSTRACT Dana (1993) ntroduced a famly of emators n te ratfed random samplng to emate te populaton mean. Followng te artcles of Dana (1993) and Kadlar and Cng (003), we propose a new famly of emators n te ratfed random samplng tat ncludes te emators suggeed by Kadlar and Cng (003), Sabbr and Gupta (005), Sng et al. (008). Up to te fr and second order of approxmatons, we obtan te mean square error (MSE) and te optmum case s dscussed. Also an emprcal udy s carred out to sow te propertes of te proposed emators. KEY WORDS Rato emator; Auxlary nformaton; Mean square error; Effcency; Stratfed random samplng. Matematcs Subject Classfcaton: Prmary 6D05 1. INTRODUCTION Te use of te auxlary nformaton n sample surveys results n consderable mprovement n te precson of emators of te populaton mean. Wenever tere s auxlary nformaton avalable, te researcers want to utlze t n te metod of ematon to obtan te mo effcent emator. Rato metod s used to obtan more effcent emates for te populaton mean by takng te advantage of te correlaton between an auxlary varable and udy varable. In te ratfed random samplng sceme, Hansen et al. (1946) and Kaur (1985) proposed rato emators usng te populaton mean of te auxlary varable. In smple random samplng, Srvaava (1971) suggeed general classes of emators wc contan many emators n te lterature. Tey prove tat te be emator n ter proposed classes s always a regresson-type emator and for te emators belongng to ts general class, no furter mprovement for ter performances s possble. Because of ts reason, Dana (199) calculated te MSE of te emators usng te k-t order approxmaton for te Taylor seres to fnd te mo effcent emator. Dana (1993) extended ts class of emators of te populaton mean wc ncludes some oter emators proposed n te lterature n te ratfed samplng and examned te MSE equaton of tese emators up to te k-t order approxmaton. Kadlar and Cng (005) extended Prasad (1989) emator, Sng and Vswakarma (006) extended Saa (1979) emator for te ratfed random samplng. Sabbr and Gupta (006) suggeed a new emator usng Bed (1996) transformaton. 010 Pakan Journal of Statcs 47

2 48 On te famly of emators of populaton mean n ratfed random samplng Sng and Vswakarma (008) suggeed a famly of emators usng transformaton n te ratfed random samplng. Moreover, Kadlar and Cng (003), Sabbr and Gupta (005), Sng et al. (008), Koyuncu and Kadlar (009) suggeed varous emators usng knowledge of populaton parameters of te auxlary nformaton n te ratfed random samplng. In ts udy, motvated by Dana (1993) and Kadlar and Cng (003), we propose a new famly of emators n te ratfed random samplng wc ncludes some new emators and some oters proposed n lterature suc as Kadlar and Cng (003), Sabbr and Gupta (005), Sng et al. (008). Consder a fnte populaton U u u u,,..., N 1 be te udy and auxlary varables assocated wt eac unt of sze N and let y and x, respectvely, u j 1,,..., N j of te populaton. et te populaton of sze, N, s ratfed nto rata wt -t ratum contanng N unts, were 1,,..., suc tat of sze n N 1 s drawn wtout replacement from te -t ratum suc tat N. A smple random sample n 1 n. et y, x denote te observed values of y and x on te -t unt of te -t ratum, were 1,,..., N and 1,,...,. Moreover, let Y 1 1 y n y n, 1 y W y 1, and N y N, Y Y WY be te sample and populaton means of y, respectvely, were W N N s te ratum wegt. Smlar expressons for x can also be defned. e y Y Y and To obtan te bas and te MSE, let us defne 0 e x X X. Usng tese notatons, 1 E e E e V, r s E x X y Y. (1.1) X Y rs r, s W r s 1 From (1.1), we can wrte were E e E e e W Sy 1 0 V 0, Y W Sxy V1,1 XY, E e, W Sx 1 1 V,0 X,

3 Koyuncu and Kadlar 49 S N 1 y 1 f n y Y N 1, and f, S N 1 x n. N x X N 1, S xy N 1 y Y x X In te ratfed random samplng, wen nformaton on an auxlary varable s avalable, a well known emator for te populaton mean s te combned rato emator defned as y RC y x N 1 X. (1.) Usng te notaton (1.1), te bas and te MSE of ts emator, to te fr-order of approxmaton, are respectvely gven by RC,0 1,1 Bas y Y V V, (1.3) RC 0,,0 1,1 MSE y Y V V V. (1.4) Dana (1993) suggeed a famly of emators for te populaton mean n te ratfed random samplng as xst xst ycst yst w 1 w X X, (1.5) were,,, and w can take fnte values. Wen tese four parameters are convenently cosen, many emators are obtanable. Also, wen one parameter s consdered as free parameter, t s possble to obtan some subclasses of emators. Here, free parameter means tat one of te four parameters s suc a scalar tat te mean square error of ycst gets te mnmum value. In oter words, we mnmze te MSE equaton accordng to te free parameter. Some emators, wc are generated from (1.5) for dfferent combnatons of,,, and w, are gven n Table 1. Rewrtng (1.5) n terms of e s, we ave ycst yst e w w e 1 1. (1.6) To obtan te bas and te MSE, te term 1 e 1 s expandable as were 3 4 e a a e a e a e a e 1... (1.7)

4 430 On te famly of emators of populaton mean n ratfed random samplng a ! 1,,3, (1.8) and smlarly te term w 1 w1 e 1 n (1.6) s expandable as were 3 4 w 1 w 1 e1 b0 b1 e1 be1 b3e1 b4e1... (1.9) b 1 w 1 w1 e1 e1 0 1,,3,...! e (1.10) Up to te fr order of approxmaton, te bas and te MSE of te emator ycst respectvely gven by I CST 1 1,1,0 Bas y Y c V c V, (1.11) I CST 0, 1,0 1 1,1 MSE y Y V c V c V, (1.1) are were jb j j0 c. Note tat te optmum value of c1 s obtaned as c V1,1 V,0. Usng ts optmal value, te mnmum MSE of te emator ycst V 1,1 MSEI mn ycst Y V0, W S y 1 c MSE ylrc (1.13) V,0 1 wc s equal to te MSE of te combned regresson emator. Ts means tat members of ts famly of emators, wc are generated by consderng one parameter free, ave te same MSE wt te combned regresson emator. To fnd te mo effcent emator among y CST, t s useful to fnd ter MSE equatons up to te second order of approxmaton. Up to te second order of approxmaton, te bas and te MSE of te emator ycst are respectvely gven by s 1 opt

5 Koyuncu and Kadlar 431 Bas y Bas y Y c V c V V c V, (1.14) II CST 1 CST,1 3 3,0 3,1 4 4,0 MSE y MSE y Y c c c V c c V c c c V II CST 1 CST 1 3 4,0 1 3, ,1 c1 c V,1 c1 c V, c1v 1,. (1.15) Te optmum value of free parameter s obtaned by mnmzng te MSE y. Soluton for te determnaton of ts value s obtaned by usng te fmnbnd functon n Matlab. To obtan te MSE equatons for te second order expressons, we ave used some results gven n Sukatme et al. (1984).. SUGGESTED FAMIY OF ESTIMATORS Followng Dana (1993) and usng some known populaton parameters, we can defne a general famly of emators for te populaton mean as were x x A B yn yst w 1 w X XA B A W A, 1 B W B. Here A and B 1 II CST, (.1) may be te populaton nformaton of te auxlary varable for te -t ratum suc as S x, coeffcent of varaton C x, skewness 1 x, kurtoss x, correlaton coeffcent xy. Note tat Koyuncu and Kadlar (009) use tese defntons n ter famly of emators.,,, and w can be defned as n Dana (1993). Many new emators, wc are generated from (.1) for dfferent combnatons of,,, and w, are gven n Table. Expressng te emator, y N, n terms of e ( = 0, 1), we can wrte (16 ) as were yn Y e e w w t e t XA XA B., (.) Up to te fr order of approxmaton, te bas and te MSE of te emator respectvely gven by N N 1,0 0, 1 1,1 Bas y Y c V c V, (.3) 1 1 1,1,0 MSE y Y c V V c V, (.4) I were c b t j j j0 j and optmum value of Ten, subtutng ts value n (.4), we can get te mnmum MSE as 1 yn are c s obtaned as c1 opt V1,1 V,0.

6 43 On te famly of emators of populaton mean n ratfed random samplng I mn N y 1 c lrc MSE y W S MSE y (.5) 1 wc s equal to (1.13). Ts means tat subclasses of ts famly ave te same MSE wt te combned regresson emator up to te fr order of approxmaton, wereas up to te second order approxmaton, we can gve MSE as II N 0, c 1 c V, c c1 c3 V4,0 c3 c1 c V3,1 1,0 1 1,1 1 3,0 1,1 1 1, MSE y Y c V V c V c c V c c V c V. (.6) To ave effcent emates, we decde to use te Kadlar and Cng (003) defnton n (.1) and by ts way we defne a new famly as were x a B yk yst w 1 w, (.7) X A B A W X A 1, B W B, a 1 W x A 1. Some new emators and suggeed emators n te ratfed random samplng lterature, wc are generated from (.7) for dfferent combnatons of,,, w, A, and B, are gven n Table 3. To obtan te MSE, we can defne a new e term as W A x X a A 1 1 A A e Expressng te emator, were t y K. (.8), n terms of e ( = 0, 1) and e 1, we can rewrte (.7) as yk Y e e w w t e, (.9) A A B Expandng terms 1 e K. 1 and w 1 w 1 te y Y e a a e a e a e a e b b t e a were te terms order of approxmaton, we can wrte n (.9), we can get b t e b t e b t e, (.10) and b are defned as (1.8) and (1.10), respectvely. Up to te fr

7 Koyuncu and Kadlar 433 yk Y Y a1 e1 e0 b1 t e1 a1e1 e0 a1b 1te1e 1 b1 te0e1. (.11) Consderng te followng equatons, s t r W A E x X y t Y r t s r A X Y s E e e e, (.1) V t s t r W A E x X y Y 1 s, t, r r we obtan te MSE of te emator, were k t X Y y K, (.13), up to te fr order of approxmaton, by MSE y Y a V V b k V a V a b k V b k V, a A I K X B. Optmum values of a1 and b1 are also obtaned as V101V00 V011V opt V00V00 V V V V V, b1 opt k V V V (.14). (.15) Subtutng tese optmum values n (.14), we can get te mnmum MSE of MSE y Y V V101V00 V00 V011 V101V011V110 I mn K 00 V00V00 V110 From (.16), we can say tat varous usages of Up to te second order of approxmaton, te MSE equaton s gven by yk as. (.16) A can affect te II K MSE y Y a V V b k V a V a b k V b k V MSE y. I mn 6 a1b a1b1 kv10 a1v 10 b1 kv01 b b1 kv01 a1 b1 3ab1 kv11 3a1b a1b1 kv11 4a1b1 kv a a V a b k V b b k V a a V a b a b k V a a V a b a b a b a b k V b b k V a a a V b b b k V a a V a a b a b k V 3 3 a1b3 a1b 1b kv130 a3v301 b3 b1 b kv031 (.17) K

8 434 On te famly of emators of populaton mean n ratfed random samplng We fnd te effcency condton for te proposed famly of emators, follows: MSE y MSE y MSE y MSE y I mn K I mn CST I mn N lrc y k, as Y V Y V V101V00 V00 V011 V101V011V 110 V1,1 00 0, V V 00V00 V 110,0 1, , V V V V V V V V V V V V (.18) If te condton (33) s satsfed, yk s more effcent tan y CST, y N, and y lrc. 3. NUMERICA EXAMPE In ts secton, we use te data concernng te number of teacers as udy varable and te number of udents as auxlary varable n bot prmary and secondary scools for 93 drcts at 6 regons (as 1: Marmara : Agean 3: Medterranean 4: Central Anatola 5: Black Sea 6: Ea and Soutea Anatola) n Turkey n 007 (Source: Mnry of Educaton, Republc of Turkey). Te summary atcs of te data are gven n Table 4. We used te Neyman allocaton for allocatng te samples to dfferent rata (Cocran, 1977). Up to te fr order of approxmaton, we compute te mnmum MSEs of Dana (1993) and suggeed famly of emators, y N, y K, usng (1.13), (.5), and (.16), respectvely. By ts way, we get te mnmum MSE of Dana (1993) and suggeed famly of emators, y, as wc s equal to te MSE of te combned N y K, as kurtoss, regresson emator, wereas we get te mnmum MSE of te suggeed emator, dfferent from te combned regresson emator. Wen we use coeffcent of varaton, skewness and correlaton coeffcent we get te mnmum MSE values as , 18.74, and , respectvely. From tese results, we can say tat te suggeed famly of emators, y, s more effcent tan te emators of Dana (1993), suggeed famly of emators, regresson emator wen we defne A K y N as correlaton coeffcent. A, and te tradtonal combned As mentoned n te prevous sectons, wen a parameter s defned as a free parameter for te famly of emators n Dana (1993) and suggeed famly of emators, y N, te mnmum MSE of tese emators s equal to te MSE of te combned regresson emator. Terefore, to nvegate te effcency among subclasses of tese famles, we decde to compute te MSE up to te second order of approxmaton. Usng (1.15), we obtan te second order MSE values of members of Dana (1993) emators, gven n Table 1, and tese MSE values are sown n Table 5. From Table 5, we observe tat ygu s te mo effcent emator for ts data set.

9 Koyuncu and Kadlar 435 Smlarly, usng (.6), we obtan te second order MSE values of members of emators, gven n Table, wt varous known populaton parameters and tese MSE values are sown n Table 6. From Table 6, we observe tat yn s te mo effcent emator for ts data set and also we note tat usng dfferent known populaton parameters of te auxlary nformaton as approxmately no effect on te MSE values. Up to te fr order of approxmaton, te emators y SD, y SK, y SD, y SD, and SK y ave te same mnmum MSE wc s equal to te combned regresson emator. Note tat y US and yr are te mo effcent emators. As seen n Table 7, up to te fr order of approxmaton, some emators ave te same MSE. For ts reason, to fnd te mo effcent emator among y K, we decde to fnd ter MSE equatons up to te second order of approxmaton by (.17). Up to te second order of approxmaton we fnd tat y US s te mo effcent emator. 4. CONCUSION In ts paper, te propertes of emators n Dana (1993) are dscussed n te ratfed random samplng and we propose two new famles of emators usng some known populaton parameters of te auxlary varable. Frly, we use Koyuncu and Kadlar (009) defnton n Dana (1993) famly of emators and we fnd tat up to te fr order of approxmaton we get te mnmum MSE equal to te MSE of te combned regresson emator and usng auxlary nformaton does not affect te MSE muc, wereas up to te second order of approxmaton t affects te MSE slgtly. Secondly, we sugge one more famly usng te defnton n Kadlar and Cng (003) and fnally, we fnd tat up to te fr order of approxmaton usng te auxlary nformaton affects te MSE enormously as seen n Table 7. Wen we defne A as a correlaton coeffcent n y K, we get te mnmum MSE as wc s qute smaller tan MSE of te combned regresson tat s Second famly of emators also ncludes some oter emators proposed by many autors n lterature suc as Kadlar and Cng (003), Sabbr and Gupta (005), Sng et al. (008). Moreover, from ts famly many new emators can also be obtaned. We examne te effect of varous transformatons of te auxlary nformaton on te famles of emators. We also udy te second order of approxmaton on te proposed famles of emators and we see tat te mnmum MSE values of te famly of emators, yk can cange accordng to defnton of y K yn A. We sow tat te proposed, can be more effcent tan te combned regresson emator. However, te MSE of te proposed famly of emators,, s equal to te MSE of te combned regresson emator suc as famly of emators n Dana (1993). y N

10 436 On te famly of emators of populaton mean n ratfed random samplng REFERENCES 1. Bed, P.K. (1996). Effcent utlzaton of auxlary nformaton at ematon age. Bometrcal Journal, 38(8), Cocran, W.G. (1977). Samplng Tecnques. Jon Wley and Sons, New-York. 3. Dana, G. (1993). A class of emators of te populaton mean n ratfed random samplng. Statca, 1, Dana, G. (199). A udy of k-t order approxmaton of some rato type rateges, Metron, 1(), Hansen, M.H., Hurwtz, W.N. and Gurney, M. (1946). Problems and metods of te sample survey of busness. J. Amer. Stat. Assoc., 41, Kadlar, C. and Cng, H. (003). Rato emator n ratfed samplng. Bometrcal Journal, 45(), Kadlar, C. and Cng, H. (005). A new emator n ratfed random samplng. Commun. n Stat.: Teory and Met., 34, Kaur, P. (1985). On te ematon of populaton mean n ratfed samplng. Bometrcal Journal, 7(1), Koyuncu, N. and Kadlar, C. (009). Rato and product emators n ratfed random samplng. J. Stat. Plann. and Infer., 139(8), Prasad, B. (1989). Some mproved rato type emators of populaton mean and rato n fnte populaton sample surveys. Commun. n Stat.: Teory and Met., 18(1), Saa, A. (1979). An effcent varant of te product and rato emators. Statca Neerlandca, 33, Sabbr, J. and Gupta, S. (005). Improved rato emators n ratfed samplng. Amer. J. Mat. and Mngt. Scen., 5, 3-4, Sabbr, J. and Gupta, S. (006). A new emator of populaton mean n ratfed samplng. Commun. n Stat.: Teory and Met., 35, Sng, H.P., Talor, R., Sng, S. and Km, J.M. (008). A modfed emator of populaton mean usng power transformaton. Statcal Papers, 49, Sng, H.P. and Vswakarma, G.K. (006). An effcent varant of te product and rato emators n ratfed random samplng. Statcs n Transton, 7(6), Sng, H.P. and Vswakarma, G.K. (008). A famly of emators of populaton mean usng auxlary nformaton n ratfed samplng. Commun. n Stat.-Teory and Met., 37, Srvaava, S.K. (1971). A generalzed emator for mean of a fnte populaton usng mult-auxlary nformaton. JASA, 66, Sukatme, P.V., Sukatme B.V., Sukatme, S. and Asok, C. (1984). Samplng teory of surveys wt applcatons. Iowa State Unversty Press, Ames, Iowa, USA.

11 Koyuncu and Kadlar 437 RW Table 1 Some members of Dana (1993) emator w c 1,..., k y y Gu w 1 y RS w ysr 0 1 ytr w ! w 1 0,... y sr w ysp yrp -1 1 MS 1 w 1 0,... 1 w 1 1 w,... y w w j ysm w j1 1

12 438 On te famly of emators of populaton mean n ratfed random samplng Table : Some members of suggeed famly of emators w c c1 1 w t 1 c w1 wt 1 wt 1 yn yn c w w w t w1 wt 1 wt 1 y N c1 1 wt 1 c 1 wt 1 wt 1 3 yn c1 1 wt 1 c 1 wt 1 yn c3 1 w t 1 w t 1 w t 1 c3 1 w t 1 c t 1 c 1 t 3 c3 1 t 6 yn c1 1 wt 1 c 1 wt 1 wt 3 c3 1 wt 1 wt yn c1 1 wt c 1 w t c 3 1 w t yn c1 1 wt c 0 c3 0 yn8-1 1 c1 1 wt 1 c 1 wt 1 wt 1 c3 1 wt 1 wt 1 yn c1 1 wt 1 c 1 w t 1 wt 3 3 c3 1 w t 1 w t yn c1 1 wt 1 c 1 w t 1 wt c3 1 w t 1 w t 1 w t 1

13 Koyuncu and Kadlar 439 Table 3: Some members of suggeed famly of emators w ysd Kadlar and Cng(003) Cx yk A B ysk Kadlar and Cng(003) yus1 Kadlar and Cng(003) x yus Kadlar and Cng(003) Cx x Cx x ysd Sabbr and Gupta (005) 0 w Cx ysk Sabbr and Gupta (005) 0 w yus1 Sabbr and Gupta (005) 0 w -1 1 x yus Sabbr and Gupta (005) 0 w -1 1 Cx x Cx x yr Sng et al.(008) x Cx yr Sng et al.(008) Cx x y C SD x y SK y US1 x y C US x x Cx x

14 440 On te famly of emators of populaton mean n ratfed random samplng Table 4: Data Statcs N 1 =17 N =117 N 3 =103 N 4 =170 N 5 =05 N 6 =01 n 1 =31 n =1 n 3 =9 n 4 =38 n 5 = n 6 =39 S y1 = S y =644.9 S y3 = S y4 = y5 S = S y6 = Y 1 = Y =413 Y 3 = Y 4 =44.66 Y 5 =67.03 Y 6 = S x1 = S x = S x3 = S x4 = S x5 = S x6 = X 1 = X = X 3 = X 4 = X 5 = X 6 = S xy1 = S xy = S xy3 = S xy4 = xy5 S = S xy6 = =0.936 = = = = =0.965 x x x = =4.593 = x4 =10.16 x5 =1.947 x6 1 x 1 = x = x3 x =3.11 x =4.084 x w 1 =0.138 =3.114 =3.748 = w =0.17 w 3 =0.11 w 4 =0.184 w 5 =0. w 6 =0.18

15 Koyuncu and Kadlar 441 Table 5: Second Order MSE (n bold) of members of te famly of emators of Dana (1993) and optmum values of free parameter MSE II yrw ygu yrs ysr ytr ysr ysp yrp yms ysm

16 44 On te famly of emators of populaton mean n ratfed random samplng t Table 6: Second Order MSE (n bold) of members of te famly of emators of 1 X W C 1 X W C x W 1 x and optmum values of free parameter x X W X W 1 1 x 1 x W C x X W1 x 1 X W1 x 1 W x 1 yn X W x 1 X W x 1 W1 x 1 y N1 y N y N 3 y N 4 y N 5 y N 6 y N 7 y N8 y N 9 y N

17 Koyuncu and Kadlar 443 Table 7: Fr and Second Order MSEs (n bold) of members of te famly of emators of y and optmum values of free parameter K MSE I MSE II ysd ysk y US y US y SD y SK y US y US yr yr y SD y SK y US y US MSE y = (wen A K mn correlaton coeffcent