(Received: March, 1988)

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1 IoUI' Soe. AI. Statllliu Yol. XLII. No. 2 (1990J.pp SAMPLING SCHEMES POVIDING UNBIASED GENERAL REGRESSION ESTIMATORS PADAM SlNGH- ad H. V. L. BATHLA-. (Received: March, 1988) SUMMARY I this paper a samplig scheme haa bee proposed for which usual regressio estimator becomes biased The proposed samplig scheme is simpler tha oe propo~cd by Sigh ad Srivastava (1980) ad i additio has the advataae of extedig it to multivariate regressio estimator ad polyomial regressio estimator. From efficiecy poit of view also the proposed samplig strategies are better tha the covetioal strategies uder use. Ke. wards: t:fficiecy; Regre,sio estimator; Mulrivariate regressio estimator; Polyomial regres~io estimator; Ratio estimator; Samplmg Scheme; Samplig strategy; Ubiased estimatio. Itroductio Sigh ad Srivastava (1980) proposed a samplig scheme for which the usual regressio estimator is ubiased. The procedure suggeskd by them ivolves calculatio of coditioal probabtlitle' if sample is to be sc:lectcd uit by uit. I this paper a alterat ive samplmg scheme bas bee sugge.;ted. The proposed samplig scheme has advatage of extesio to polyomial regressio estimator ad muitivar,iate regressio estimators. The efficiecies of the proposed samplig strategies with those i vogue have also bee compared. -Istitute for Researeh i Medical Statistics ICMR). New Delhi. "Idia A8ricultural Statistics Research Istitute, New Delhi.

2 GIlHIiRAL RIlGRB~8ION ESHMA10R Proposed Samplig Scheme. 2.1 Samplig Scheme Providig Ubiased Regressioll Estimator Suppose that the populatio uder cosideratio cosists of N distict ad idetifiable uits ad a ~amole of size is desired to be draw from it. Further. suppose y i~ the study va;-iable ad x il the auxiliary variable. We as~ume that the iformatio o the auxiliarv variable is available for all the uit, of populatio The samplig schemc= providig u biased regressio e~timator is as uder: (a) Select a sample of size bv simple radom samplig ad without replacemet Calculate 3 2 (the sample mea square) for the sample 0: so selected, where I s!= --~ ~ (x, - x)', lt = ~ X, x - i= 1 i = 1 (b) Select a radom umber j betwee 0 ad M where M"> Max (s~) It is importat to metio here that maximum of s! will geerally be kow sice x~s are all kow to u~. (c) Ifj is less tha or equal to s; relai the sample selected at step (a) otherwise proceed to step (a) agai for selectig a sample ad cotiue tho process till the sample is ultimately selected. It is ealy to see that for abovo lamolig scheme the probability of selectig a sample's' is proportioal to tbe variability i tbe sample for the auxiliary variable i.e. ft P,= i s_ (~) S; It is assumed here that o combiatio of uits from N uits of the populatio have Rame value of the auxiliary variable so that all samples have o zero probabilities of selectio This samplig scheme is simp ler tba that by Sigh ad Srivastava ( 980) i the sese that it does ot ivolve calculatio of coditioal probability of succe.1s at the secod draw. However it ivolves umber of rejectios. But because 001y oe sample is to be selected it may ot be a limitatio.

3 I 208 JOURNAL O. THB INDIAN SOCIBTY OF AGRICULTUltAL ITAllSTICI We, ow, exted the proposed samplig scheme for makig polyomial regressio estimator as well as multivariate regressio estimator ubiased i the followig sectios. 2.2 Samplig Scheme Providig Ubial,d Polyomial R,gressio Estimator We shall cosider the samplig scheme for which quadratic regressio estimator is ubiased. The extesio for a geeral kth degree polyo mial regressio estimator is straight forward. The proposed samplig Icheme cosists of the followig steps. (a) Select a sample of size by simple radom samplig without replacemet. For the selected sample, calculate where m =. l: (XI - x)jl (Yt - ')Il 1=1 (b) Select a radom umber j from 0 to M, where, M;;;' Max (A.) Here also M will be kow geerally sice xc are all kow to UI. However, for the extreme cases M ca safely be take as, (~) [llu ll" -!l~o -!A o J (c) if j is leas tha or equal to Aj retai the sample otherwise proceed to step (a) for lelectil aother samplo ad cotiue the process till a sample is ultimately selected. For this samplig scheme the probability of selectig a sample is a-ived by

4 OBNBRAL kb,ou.sion I:STlMATOas where I (Z) T= (N) 2: (mlo m.. - m!o - mlol s-1 The estimator to be used i this situatio is,: =, - b l.i' - hi.i2', where.ii, x are the meas of (x') for sample ad populatio, x' = % - X. ~I' = :;. - XI ad b ad b. l are the regressio coefficiets of the quadratic regressio. give by ad 2 me. mh + 2 mao mil Jt - m20 mil - mi. mil - 2% mi. mil m m,. - m~ - mlo Now t: =,- m'l mil mil m % - mlo mil - m mit - 2% mu m., a' m ZI m&o - mt'o - m.8 mil mil - m mil. %.' -mllm60 - m~ - m;o ad EO:) = 1 (~)T ~ [t:j (mil m" - m:o - mio) SfS = ~ ~ [, (m m.. - m~ - m 3 3) - (m" "' + 2 mae mll X - m:o ml1 - m,. mu - 21 mal mil) I' - (m mal - mid mil),il'] where ~ doates the average over (If) samples of the samplo space S.

5 210 JOUUUL OF THIlINDIAN SOCIIlIY OF AORICULTUR.AL ITAllSCI T 1 ~ [ (y + f) { 1: (x, - of)' 1: (x, - x)' r=1 r=1 ' - ( 1: (x, - x)' ) - ( 1: (x, - X)3)'}, =1, ~ (X, - X)4 1: (X, - ox ) (y, - ') {, = 1 r - l + 2 1: (x, - "X) (y, - y) - 1: (x, - X)3,,,,,1 r=1 1: (X, -X),. (y, - Y), = 1-2 (X + x') 1: (x, - x)' ~ (X, - X)2 (y, - y) } X' r = 1, = 1 _ { ~ (X, - X)2 ~ (X, - X)2 (y, - y) r=1,=1 1: (X, - X)I. 1: r = 1 r = 1-1 Y + N (N - l)t N N N N N 1: X; Y~ + 2 1: X~' 1=1 ;=1 l:. X ~ Y; (X + X') - ( 1: X;, 1:_ Y; Xi 1=1 1=1 ' ; = 1 N N 'N 1:: X;I 1: X;I Y~ - 2 (X + x') 1:: X':. ; = I ' i =0' 1 ' 1=,1_ N N N N. '} " ~ x~ - { ; X'2- l:. X'~Y; _ ~ 1:: X~8 1:: X, Y, ; = 1 ; = 1 ';=1 ;=-1 ;-1 ~ X;' ] ; = 1 =y )2

6 OINBI.AL UORBISIOM ESTIMATORS 211 Therefore, the usual quadratic regressio estimator is ubiased for this sample scheme. The variace of this estimator to the first order of approximatio is give by N-,,(N - J) (L01 (1 - l)~2» where "/)(:) is the correlatio ratio. Thus for larg~ samples the efficiecy of the proposed samplig scheme is same as the efficiecy of the usual quadratic regressio estimator uder simple radom samplig without replacemet. 2.3 Sampli"g Scheme Providig U"bialed Multi-variate RegreJJio Eltimator Here we cosider the samplig scheme for which 2-variate regressio estimator is ubiased ad the extesio for a geeral k-variate regressio estimator thereafter is straightforward. Let Y., Xl' ad X,/ be the values of the study character (y), auxilliary characters Xl ad XI for ith uit of the populatio (i= 1, 2,... N). It is assumed that the auxilliary characters are kow for all the populatio uits. The samplig scheme the cosists of the followig steps: (a) Select a samale of size " by simple radom samplig without replacemet. For the selected sample calculate (b) Selcct a radom umber j from 0 to M, where, M;> Max. (L.) (c) If j ~ L., retai the sample, otherwise proceed to step (a) for selectig aother lample ad cotiue the process till a sample is selected. Now, it i. clear thai for the suggested samplig scheme, the probabi Jity of selectig a sample 's' is

7 212 10U&NAL OP THB I.DIAN SOCieTY OP AORICULTURAL ITATISTICS where, Udcr thc suggested samplig scheme the estimator is t;' =, - b l.11 - b l.t;whcre hl ad b 2 arc partial regressio coefficiets. Now..

8 GBNSRAL IUIORSISION ESTIMATOIlI 213 This o simplificatio reduees to + ;- ~ [t{ i: xl~ E x;~ - (.;. x;, XI' )I} 1 r=l r _ 1 r { 1: r = 1 XI', 'I =y Therefore, the ulual 2 - variate regressio estimator is ubiased for this samplig scheme ad the variace of this estimator to the first order of approximatio is give by where R~.2J represet. the multiple correlatio coefficiet of y with Xl ad.1'2' The proposed samplig.chemes together with the estimato will hereafter be referred al samplig strategies. 3. CODipari.o. For large samples, it has bee observed that the efficiecy of the usual estimators uder proposed samplig scheme. is of the same order as that of the same estimators uder simple radom samplig without replacemet. _For comparig the efficiecy of the proposed strategies for small lamples, we cosider the followig empirical comparisos. 3.1 We colider 18 populatios of size 15 each. These populatios

9 214 JOURNAL O' THB INDIAN SOCIBTY O' AGRICU1.TUkA1. ITATISTICS have bee geerated satisfyig the model, The value of g are 0, 1 ad 2 with differet combiatios of :x, (31 ad ~" The samplig strategies cosidered for compariso are the followig: Quadratic regreasio estimatior uder the proposed samplig scheme-2. 2 Ratio estimator uder Midzuo-Se samplig. 3. Regressio estimator uder simple radom samplig without replacemet. 4 Ratio estimator uder simple radom samplig without replacemet. 5 Simple mea uder simple radom samplig without replacemet. It ca be see that strategies I ad 2 provide ubiased estimators whereas, the remaiig strategies provide biased estimators. The variaces/mea square errors of above strategies for sample of size 4 have bee worked out as deoted by Vt. Vt. V 3, v, ad V6 ad pre<eted i Table 1. Followig observatios ca be made about the efficiecy of differet samplig strategies. Populatios 4 to 9, have bee geerated whe the fuctioal relatioship betwee y ad x i~ liear ad for such cases, the suggested samplig Icheme-2 il better tha the usual regressio estimator but the differece i variace/mea square error is of a lower order. I situatios, where the relatioship betwee y ad x is other tha liear, e.g. populatios 1 to 3 ad 10 to IS, the gai i efficiecy of the suggested samplig scheme-2 over other strategies is quite cosiderable. These results idicate that the performace of the samplig scheme 2 is highly satisfactory. particularly whe the fuctioal relatioship betwee y ad x is other tha liear. 3.2 Here we cosider 18 populatios of size 12 each. These have bee geerated satisfyig the model

TABLE I-VARIANCES/MEAN SQUARE ERRORS OF DIFFERENT SAMPLING STRATEGIES 0!'I:I FOR = " :II:!'I:I Populatio VI Vs V V, Vs 8 1. 0.1546 X 10' 0.1146 X log 0.7120 X 10i 0.1371 x los ".5972 X 10' "... IlII 2.

10 TABLE I-VARIANCES/MEAN SQUARE ERRORS OF DIFFERENT SAMPLING STRATEGIES 0!'I:I FOR = " :II:!'I:I Populatio VI Vs V V, Vs X 10' X log X 10i x los ".5972 X 10' "... IlII X X X X X log ( x 10' x los X X los X 10 8 '2: IlII X ]()O x x 10' x 10' X x 10' X b34 X 10' X X 10 2 ~ X 10' x 10' X 19' 0. ]463 X ]0' X 10' 7., x 10' X 10' x los O X 10' x 10.. I X 10 1 U. ~049 X 10' X 10' X X 1() x 10' X loa X 10' X 10' X x WI X X ItS X 10' X 10' >< 10' x 10' x lo X x 10' x 10' x X 10' X X log X X 10 i X X 10 ~ X H is 14. 0,1889 X 10' 0.28 S0 X X X 10" X 106 IS x 10' x 10' X 10' 0.:411 X X 10' >'" t"',. rtl > X 10' X 106 O. 17fO X X los X log X ]01 0.Il45 X X lo r, X X ]889 X 10' ; X X X IC X 10 6 N YO

TABLE 2-VARIA~(fS / MFAN ~QUARE ERRORS OF DIFFERENT SAMPLJ1'iG STRA1EGIES FOR = 4 t-.) 0- Populatio VI V. V. V, P6 Va... 0 c:: ]. 2. 7. 0.1630 x 10' 0.4949 X 10 1 0.1630 x 10 0.5033 X 10 1 0.

11 TABLE 2-VARIA~(fS / MFAN ~QUARE ERRORS OF DIFFERENT SAMPLJ1'iG STRA1EGIES FOR = 4 t-.) 0- Populatio VI V. V. V, P6 Va... 0 c:: ] x 10' X x X x JO~ 0.]649 X X 1( X X ~07] X X :625 X 10' X It' X JOG X ~09 X x ]863 X ] X X X (0 X lcol 0.<3:0 X 1(,0 0:077 X 1(, X X x ] X X ]0 1 0.]H2 X ]0' X X 10 0 == r!i S X X 10'! 0,7828 X X IC' X X 10 1 i x 10' X l34 X 1(,3 0]217 X J28 X X ]0' t:i-> II '6 X H.' 0.]630 X H;o X 10' 0.4]56 x lo' X ] X ] (, x ](,3 0.16:0 x )(., x ]( X X X l(ol X 1(4 0.2?34 X 10' X X :;758 X ic 4 04Ct8 X 1(i x 1( 0.14: 5 x ] x 1( 0. 99~6 X JOI x ](" o ]C fa X J X J X 10' 0.~071 X ]0 1 09~1? X ] X 10 O.SS~O x ] X :0 x ](a x <16 X lc X ] x 10' 0.1U.9 X 10" X X 10' X 10' 0.'1813 X lc X X 10' X X ] X lq1 0. 3~03 X X X 1(J X ~9 X X X 10' X X ]0' X X X 10' X X X ]0 3 G.5113 X X X III 21 > I"" rjt 0 a; o.i 0< 0.. '"- > 0 c:: I"" ooi c: '"> I"".... > ooi I: rill

BNERAL RBOREI!ISIO~ EITIMATORS 217 The strategies cosidered for compariso are as follows: 1. Suggested samplig strategy-3. 2. Ratio estimator uder Midzuo-Se samplig scheme. 3.

12 BNERAL RBOREI!ISIO~ EITIMATORS 217 The strategies cosidered for compariso are as follows: 1. Suggested samplig strategy Ratio estimator uder Midzuo-Se samplig scheme. 3. Regressio estimator uder simple radom samplig without replacemet. 4. Ratio estimator uder simple radom samplig without replacemet. S. Simple mea uder simple radom samplig without replacemet. 6. Suggested samplig strategy-i. I strategies (2), (3), (4) ad (6) the auxiliary variable used i. Xl oly. Estimators of populatio mea or total obtaied from (1). (2), (5) ad (6) are ubiased, whereas, (3) ad (4) provide biased estimators. The variaces/mea square errors of the above strategies deoted by VI> V" Va, VI' Va ad V, for sample of size 4 have bee computed ad results preseted i Table 2. The followig observatios ca be made from the above compariso: (i) The suggested samplig strategy 3 is better tha the usual regressio estimator i all the populatios. (ii) The suggested strategy 3 is better tha the ratio estimator uder Midzuo Se samplig scheme i all the populatios. These results idicate that the performace of the suggested samplig strategy-3 i. highly satisfactory ad the gai i efficiecy is quite cosiderable. REFERENCE Sigh, P. ad Srivaltava. A.K. (1980): Sampli, scheme providig ubiased regressio estimator. Biometrika S-209.

Chapter 8. Confidence Interval Estimation. Copyright 2015, 2012, 2009 Pearson Education, Inc. Chapter 8, Slide 1

Chapter 8. Confidence Interval Estimation. Copyright 2015, 2012, 2009 Pearson Education, Inc. Chapter 8, Slide 1 Chapter 8 Cofidece Iterval Estimatio Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 1 Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for