Sampling Theory MODULE V LECTURE - 14 RATIO AND PRODUCT METHODS OF ESTIMATION

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1 Samplg Theor MODULE V LECTUE - 4 ATIO AND PODUCT METHODS OF ESTIMATION D. SHALABH DEPATMENT OF MATHEMATICS AND STATISTICS INDIAN INSTITUTE OF TECHNOLOG KANPU

2 A mportat objectve a statstcal estmato procedure s to obta the estmators of parameters of terest wth more precso. It s also well uderstood that corporato of more formato the estmato procedure elds better estmators, provded the formato s vald ad proper. Use of such aular formato s made through the rato method of estmato to obta a mproved estmator of populato mea. I rato method of estmato, aular formato o a varable s avalable whch s learl related to the varable uder stud ad s utlzed to estmate the populato mea. Let be the varable uder stud ad be a aular varable whch s correlated wth. The observato o ad ) o are obtaed for each samplg ut. The populato mea of (or equvaletl the populato al must be kow. For eample, ' s ma be the values of ' s from some earler completed cesus, some earler surves, some characterstc o whch t s eas to obta formato etc. For eample, f s the quatt of fruts produced the th plot, the ca be the area of th plot or the producto of frut the same plot prevous ear.

3 Let (, ),(, ),...,(, ) be the radom sample of sze o pared varable (, ) draw, preferabl b SSWO, from a populato of sze N. The rato estmate of populato mea s = = assumg the populato mea s kow. The rato estmator of populato al s ( ) N = N = where s the populato al of whch s assumed to be kow, ad are the = = sample als of ad respectvel. The ( ) = =. ( ) ca be equvaletl epressed as Lookg at the structure of rato estmators, ote that the rato method estmates the relatve chage that occurred after (, ) were observed. It s clear that f the varato amog the values of s earl same for all =,,..., the values of (or equvaletl ) var lttle from sample to sample ad rato estmate wll be of hgh precso. = = = = = 3

4 Bas ad mea squared error of rato estmator: Assume that the radom sample (, ), =,,..., s draw b SSWO ad populato mea s kow. The E ( ) = N N = ( geeral). Moreover t s dffcult to fd the eact epresso for follows: Let ε0 = = ( + εo) ε = = ( + ε). E ad E. So we appromate them ad proceed as Sce SSWO s beg followed, so E( ε ) = 0 0 E( ε ) = 0 4

5 E( ε ) = E( ) 0 N = S N f S = = f C where N S f =, S = ( ) C = N N N ad = s the coeffcet of varato related to. Smlarl, f E( ε ) = C E( εε 0 ) = E[( )( )] N N = ( )( ) N N f = S f = ρss f S S = ρ f = ρcc = 5

6 S where C s the coeffcet of varato related to ad s the correlato coeffcet betwee ad. = ρ ε ' s, Wrtg terms of we get = ( + ε 0) = ( + ε ) ( ε0)( ε). = + + Assumg ε <, the term ( + ε ) ma be epaded as a fte seres ad t would be coverget. Such assumpto meas that <,.e., possble estmate of populato mea les betwee 0 ad, Ths s lkel to hold true f the varato s ot large. I order to esures that varato s small, assume that the sample sze t s farl large. Wth ths assumpto, = ( + ε )( ε + ε...) 0 = ( + ε ε + ε εε +...). So the estmato error of 0 0 = ( ε ε + ε εε +...). s 0 0 I case, whe sample sze s large, the ε0ad ε are lkel to be small quattes ad so the terms volvg secod ad hgher powers of ε ad ε would be eglgbl small. 0 6

7 I such a case ( ε ε ) ad E ( ) = 0. 0 So the rato estmator s a ubased estmator of populato mea upto the frst order of appromato. If we assume that ol terms of ε0ad ε volvg powers more tha two are eglgbl small (whch s more realstc tha assumg that powers more tha oe are eglgbl small), the the estmato error of ca be appromated as ( ε ε + ε εε ) 0 0 ad f f E ( ) = 0 0+ C ρcc ( ) ( f Bas = E ) = C( C ρc) upto secod order of appromato, the bas geerall decreases as the sample sze grows large. 7

8 The bas of s zero,.e., Bas( ) = 0 f E( ε εε) = 0 0 Var( ) Cov(, ) or f = 0 or f Var( ) Cov(, ) = 0 Cov(, ) or f Var( ) = 0 ( assumg 0) Cov(, ) or f = = Var ( ) whch s satsfed whe the regresso le of o passes through org. Now, to fd the mea squared error, cosder MSE ( ) = E ( ) E = E ( ε0 ε+ ε εε 0+...) ( ε + ε εε). 0 0 Uder the assumpto ε < ad the terms of ε0 ad ε volvg powers more tha two are eglgble small, f f f MSE( ) = C + C ρcc f = C + C ρcc up to the secod order of appromato. 8

Comparison of Dual to Ratio-Cum-Product Estimators of Population Mean

Comparison of Dual to Ratio-Cum-Product Estimators of Population Mean Research Joural of Mathematcal ad Statstcal Sceces ISS 30 6047 Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Comparso of Dual to Rato-Cum-Product Estmators of Populato Mea Abstract