Two phase stratified sampling with ratio and regression methods of estimation
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1 CHAPTER - IV Two phase stratified samplig with ratio ad regressio methods of estimatio 4.1 Itroductio I sample survey a survey sampler might like to use a size variable x either (i) for stratificatio or (ii) for icorporatio i estimatio procedure or (iii) for selectig a sample. Sometimes oe might thik of usig x both for (i) ad (ii) or for (i) ad (iii). I this chapter we cosider the situatio whe a auxiliary variable x is used both for stratificatio ad for ratio or regressio method of estimatio. Let the fiite populatio JJ of size N cosists of L strata of sizes NJt N2, ;Nl with Nh uits belogig to /t-th stratum. Whe the sizes of strata are ot kow, a iitial SRSWOR sample Sj of fixed size! is selected ad the classified ito differet strata with [ uits fallig i the h-th stratum i. slh (h = 1,2,..,L) with T,h = '. I the secod phase a SRSWOR sample of size hf/ is draw from slh of size idepedetly of each h to observe the mai variable y. We assume that! is so large that 77,( > 0 for each h. We also assume that at the secod phase a costat proportio of uits iitial sample. gh = 'flll are sampled from the h-th stratum of the,v 0 (i ' > V* ' * ' - :' C
2 Ratio method estimatio Let us defie a ratio estimator uder two phase stratified samplig. L -// ~ yh / wh^xh h (4.2.1) where yhf/, xhf/ are the sample mea of the h th stramm based o a sample of size,fh; x/h is the sample mea of the hth stratum based o a sample of size 'h wi, = h/' Theorem 4.1 Uder two phase stratified radom samplig yis approximately a ubiased estimator of Y for large value of u Proof: E(y«*) W h -II Yh -/ L -// = Ei 2u wh xh E2 h=l II -/ /. \ r/' +'o v."»// / (4.2.2) where, = A^/ZV = W/r ad is based o with 2?(yj[) = -
3 76 Theorem 4.2 If the first sample is a radom sample of size /, the secod sample is a radom stratified sample from the first, with fixed gh (0 < gh < 1), the = 1 N \ I 4+E f-l-i) / a=i k ; (4.2.3) where S,h = 4 + Rl 4-2 Rh ; - t N Sy - K yrj) syh» sxh are populatio variaces of y ad x for the /z-th stramm respectively ad 5 A is the populatio covariace betwee x ad y for the /2-th stratum. Proof = Eiv2(%) + ViE2(yst) (4.2.4) Now,,^(5*,) a=i // yh -/ Xh,I>* : h=l / -// )2 yh -/ -/ // xh~yh / = ie w* a=i // / h h /2 v v 2 1?rA = Wh-~ a=i w. k?/2 VA where s = s + R? 4 2Rh 4a
4 77 Syh, are the variace based o sampled h uits i the iitial sample of the h-xh stratum ad s^,h is the covariace based o sampled 'h uits i the iitial sample of the h-th stratum ad r[ = y!h I x[. E1V2(yRJj= L> h=1 u-l) whsl it f--1] [8k ) h=l ; (4.2.5) L -H L ad VlE2$to) = VlE2 Y>kqfxll = Wkyl h-1 Xl < N S; (4.2.6) / From (4.2.4), (4.2.5) ad (4.2.6) we fid F(yJ = = (l i) L S? + E U - l) lv N) y *-i k8* > 2 wl h ^rh s, Theorem 4.3 A ubiased estimator of is N' N-1 f -lh Sk -1 hsrh + N- '-\ Zyiyl-'ylft=l ghj*= 1 J (4.2.7) where srh2 = sy2 + Rh//2 - IR^s^, yhj is the j-th observatio of /?-th stramm ad = yf / Jf.
5 78 Proof L Nh Est.(N-l)Sy = Est.'E y -MyL - h=l j=l (4.2.8) ad w E-*Etf *=i h y-i L If, TjEE^i iv A-l ;=1 (4.2.9) Form (4.2.8) ad (4.2.9) st(7v-l)s 2 = TV,, //.. Hl J=1 It ca be easily see that Est-i h=l (1-1 wa _ (J-i) l J ; h=i v ) WhSrh (4.2.10) (4.2.11) From (4.2.10) ad (4.2.11), we write ±_1 r N N N-l *=i i! j=i r\ a. w Hhc. M... i 1 + E (gk -1 \ 2 ^ Vrh (4.2.: Hece the result.
6 Optimum allocatio Cosider the cost fuctio C = C'' + 'ECkh ( ) h = l where d = Cost per uit i the first phase sample; d1 h = Cost per uit i the secod phase sample. Sice f,h is a radom variable, the expected cost is E(C) = C1 (stf) = C ' +» 'Y, Ch8hWh h=i ( ) because " = hgk, = ghe['h) =!Whgh. The product C* r(y**h- y N w, S,2 + E / 1 \ -l WH& h=l is miimised if ad oly if C' c[g>wh 2 J WtS^ s; - E A-1 ( ) This gives optimum value of as = A SmVc7 s,2.-e^4,// ( )
7 80 Hece, the optimum variace is 2 v( yjjopt = c* ^ ( ) N 4.3. Regressio method of estimatio samplig. Let us defie a regressio estimator uder two phase stratified radom L h=1 (4.3.1) where /3h is the kow populatio regressio coefficiet for the /i-th stratum. Theorem 4.4 Uder two phase stratified radom samplig yreg_st is a ubiased estimator of Y. Proof L L (4.3.2) where is the sample mea of the h th stratum based o a sample of size ^.
8 81 Theorem 4.5 If the first sample is a radom sample of size 1 ad the secod sample is a radom stratified sample from the first with fixed gh (0 < gh < 1), the V(y Reg-st) ( '~ N \ i. l-i WhS^ M- (4.3.3) where Sxh2 is the populatio variace of y for the /z-th stratum ad ph is the populatio correlatio coefficiet betwee x ad y for the /z-th stratum. Proof y(y^) = + (4.3.4) Now E^y^) r // o /-/ //v Ja + Pa(*a~*a ) = iewaf2(^/-ma/) / = ie w* II / V «A»A ) + p*42-2p p*v4) L M W.S, (i-p ^,8h, Tt -E (4.3.5) ad ^(7**-*) = W*K +?h(.4-*h)
9 82 Fi vl = h=l J l ' N (4.3.6) That completes the proof. Theorem 4.6 A estimator of is ^(y Reg-st) N1 N-l^t '-\h~ i -1 ^ Sh j (i-ps)^ TiT 1\L 1 "* N~ ' J\-\ 1 2 /= L Vkj - y.reg-*\ -1 *=i Sr y=i (4.3.7) where ph is the estimated value of ph based o l Proof: L Nt Writig (IV-1)5^ = E E yl- Ny2,, h=l j=1 we ca see that it has a ubiased estimate If N _ // 'r r 2 _/=* ^ // ^5 ^ " y=l (4.3.8) Also sr. I /i=i 21 WVy, -1 M) &h II' Result follows from (4.3.7), (4.3.8) ad (4.3.9). A: f',.x /cjl P.tivWcf'' (4.3.9)
10 Optimum allocatio Cosiderig cost fuctio i ( ) the optimum value of the variace is obtaied by miimisig V(y ) + JL 'jreg-st' jy ( c'+t.c;gkwh h=1 X r, l se+r / /i=i ^ gh ) (l-p with respect to gh ad it exists if ad oly if c' cl'ghwh h=1 Sh Hece the optimum value of gh is 8h = \ ^-E^-pIKs h=1.// ( ) ad hece the optimum value of ' ca be obtaied by the expected cost ad the substitutig the optimum values of ' ad gh, the optimum value of the variace is obtaied as Numerical illustratio Cosider a data collected i a complete eumeratio of 256 commercial peach orchards i North Carolia i Jue 1946 (Fiker, 1950). Here the area
11 84 is divided geographically ito three strata. The umber of peach trees i a orchard is deoted by xhi ad the estimated productio i bushels of peaches by Yhi- Strata wh Syh2 Q 2 &xh Syxh xh Yh Srh2 Ph Let the expected cost of the experimet be C*=50 ad the cost for each uit of the sample at the secod phase be cf=0.5. Hece for SRS, a sample of size =100 is permissible. Now, cz _ V rar q* &y ^ yy h ^yh h=l h=1 ' Hece, fl_v U atj From ( ) we have S* = , sice N=256 2 h = 1 N { VC ) / 50}
12 85 We fid < V(yra) if d < ad hece further takig the cost for each uit of the sample at the iitial stage, C/=0.15 we fid = Also from ( ) we have Rst'opt N -E (1 -pfjwhs^c'+ewhsyh y(l-p^)c 50 y {c yjc^ = The relative precisio of the various methods ca be summarized as follows: Table 4-1 Samplig Method Method of Estimatio Relative Precisio (%) 1. Simple radom Mea per uit Stratified radom Two phase Stratified radom Two phase ratio Stratified radom Two phase regressio
13 Determiatio of sample size Further, from ( ) the optimum values of samplig fractios are: gj = , g2 = ad g3 = ad hece from the expected cost give by ( ) we fid *'=178, E(")=6, E{'l)=15, (f) =25 Also from ( ), the optimum values of samplig fractios are: gj = , g2 = ad g3 = ad hece from the expected cost give by ( ) we fid /1M80, E(l!)=6, («") =15, (f) = 25 Summary ad Coclusio I this chapter a attempt has bee made to costruct ratio ad regressio estimators uder two phase stratified radom samplig i presece of oe auxiliary variable. Numerical illustratio shows that the regressio estimator uder two phase stratified samplig perform better i terms of efficiecy with respect to other competitive estimators.
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