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1 A Geeral Family of Estimators for Estimatig Populatio Mea i Systematic Samplig Usig Auxiliary Iformatio i the Presece of Missig Observatios Maoj K. Chaudhary, Sachi Malik, Jayat Sigh ad Rajesh Sigh Departmet of Statistics, Baaras Hidu Uiversity Varaasi005, Idia Departmet of Statistics, Rajastha Uiversity, Jaipur, Idia Correspodig author Abstract This paper proposes a geeral family of estimators for estimatig the populatio mea i systematic samplig i the presece of orespose adaptig the family of estimators proposed by Khoshevisa et al. (007). I this paper we have discussed the geeral properties of the proposed family icludig optimum property. The results have bee illustrated umerically by takig a empirical populatio cosidered i the literature. Keywords: Family of estimators, Auxiliary iformatio, Mea square error, Norespose, Systematic samplig.. Itroductio The method of systematic samplig, first studied by Madow ad Madow (944), is used widely i surveys of fiite populatios. Whe properly applied, the methods pocks up ay obvious or hidde stratificatio i the populatio ad thus ca be more precise tha radom samplig. I additio, systematic samplig is implemeted easily, thus reducig costs. I this variat of radom samplig, oly the first uit of the sample is selected at radom from the populatio. The subsequet uits are the selected by followig some defiite rule. Systematic samplig has bee cosidered i detail by Cochra (946) ad Lahiri (954). Reviews of the work doe i the field have bee give by ates (948) ad
2 Bucklad (95). The applicatio of systematic samplig to forest surveys has bee illustrated by Hasel (94), Fiey (948) ad Nair ad Bhargava (95). Use of systematic samplig i estimatig catch of fish has bee demostrated by Sukhatme et al. (958). The use of auxiliary iformatio has bee permeated the importat role to improve the efficiecy of the estimator. Kushwaha ad Sigh (989) suggested a class of almost ubiased ratio ad product type estimators for estimatig the populatio mea usig jackkife techique iitiated by Queouille (956). Afterward Baarasi et al. (993) ad Sigh ad Sigh (998) have proposed the estimators of populatio mea usig auxiliary iformatio i systematic samplig. Khoshevisa et al. (007) suggested a geeral family of estimators for estimatig the populatios mea usig kow values of some populatio parameters i simple radom samplig, give by a + b t = y (.) α(ax + b) + ( α)(a + b) where y ad x are the sample meas of study ad auxiliary variables respectively. is the populatio mea of auxiliary variable. a 0 ad b are either real umbers or fuctios of kow parameters of auxiliary variable. α ad g are the real umbers which are to be determied. Here we would like to metio that the choice of the estimator depeds o the availability ad values of the various parameter(s) used (for choice of the parameters a ad b refer to Sigh et al. (008) ad Sigh ad Kumar(0)). I this paper we have proposed a geeral family of estimators for estimatig the populatio mea i systematic samplig usig auxiliary iformatio i the presece of orespose followig Khoshevisa et al. (007). We have also derived the expressios for miimum mea square errors (MSE) of the family with respect to α. A comparative study is also carried out to compare the optimum estimators of the family with respect to usual mea estimator with the help of umerical data. g. Proposed Family of Estimators Let us suppose that a populatio cosists of N uits umbered from to N i some order ad a sample of size is to be draw such that N = k ( k is a iteger). Thus
3 there will be k samples each of uits ad we select oe sample from the set of k samples. Let ad be the study ad auxiliary variable with respective meas ad. Let us cosider yij(xij) be the th j observatio i the th i systematic sample uder study (auxiliary) variable ( i =...k : j =... ). Wwe assume that the orespose is observed oly o study variable ad auxiliary variable is free from orespose. Usig HaseHurwitz (946) techique of subsamplig of orespodets, the estimator of populatio mea, ca be defied as where y ad y y yh = (.) + y h are, respectively the meas based o respodet uits from the systematic sample of uits ad subsample of h uits selected from o respodet uits i the systematic sample. The estimator of populatio mea of auxiliary variable based o the systematic sample of size uits, is give by x ij j= x = ( i =... k ) (.) Obviously, y ad x are ubiased estimators. The variace expressio for the estimators ad ( x) where y ad x are, respectively, give by N V y = L { + ρ} S + WS V = { + ( ) ρ } S (.3) (.4) ρ ad ρ are the correlatio coefficiets betwee a pair of uits withi the systematic sample for the study ad auxiliary variables respectively. S ad respectively the mea squares of the etire group for study ad auxiliary variable. S are be the mea square of orespose group uder study variable, W is the orespose rate i the populatio ad L =. h S
4 Let us assume that the populatio mea is kow. Thus the usual ratio ad product estimators of the populatio mea uder orespose based o a systematic sample of size, ca be respectively defied as ad y y R = (.5) x y P = y x (.6) To obtai the biases ad mea square errors, we use large sample approximatio. y = ( + ) e 0 x = ( + ) e e such that E ( e 0 ) = ( ) ( ) E e 0 = ( ) e V y V( x) E = ad E ( e 0 e ) = where respectively. E = 0, ad L S = { + ρ } C + W, = { + } C, Cov y, x ρ = { + ρ } { + ρ} ρcc C ad C are the coefficiets of variatio of study ad auxiliary variables Expressig the equatios (.5) ad (.6) i terms of i expectatios the bias expressios of the estimators of by ad y R B = + y P { ρ}( Kρ ) C B = { + ρ} Kρ C e s ( 0,) i = ad takig y R ad y P, are respectively give (.7) (.8)
5 where, ρ = { + ρ} { + ρ } C ad K = ρ. C The mea square errors (MSE s) of y R N MSE = + ad P y MSE = + y R ad y P, are respectively, give by + ρ L C K C + W S { } ( ) ρ ρ N { } ( ) ρ ρ C + + Kρ C + L W S (.9) (.0) Motivated by Khoshevisa et al. (007), we ow defie a family of estimators of populatio mea based o a systematic sample of size i the presece of orespose as t g a + b = y (.) α( ax + b) + ( α)( a + b) This family ca geerate the orespose versios of a umber of estimators of populatio mea icludig the usual ratio ad product estimators o differet choices of a, b, α ad g.. Properties of Expressig t t a where λ =. a + b t i terms of e i s, we get ( + e )( + αλe ) g = y 0 (.) We assume that λ e < so that the right had side of the equatio (.) is expadable i terms of power series. Expadig the right had side of the equatio (.) ad eglectig the terms i e i s havig power greater tha two, we have
6 g(g + ) t = e0 gαλe + α λ e gαλe0e (.3) Takig expectatio both the sides of equatio (.3), we get the bias of t up to the first order of approximatio, as ( t ) B = { + ρ } ( g + ) N g C α λ gαλkρ (.4) Squarig both the sides of the equatio (.3) ad the takig the expectatio, we obtai the MSE of t up to the first order of approximatio, as ( t ) N MSE = +. Optimum Choice of α { } ( ) ρ ρ C + g α λ gαλρ K C I order to obtai the miimum MSE of respect to α ad equatig the derivative to zero, we get { + ρ }[ αg λ gλρ K] C L + ( ) t, we differetiate the MSE of The equatio (.6) provides the optimum values of α as W S (.5) t with = 0 (.6) ρ K α = gλ (.7) Puttig the optimum value of α from equatio (.7) ito the equatio (.5), we get the miimum MSE of t, as ( t ) mi MSE = + { ρ }[ C K C ] ρ L + ( ) W S (.8)
7 The miimum MSE of t, is same as the mea square error of the usual regressio estimator i systematic samplig uder orespose. 3. Empirical Study For umerical illustratio, we have cosidered the data give i Murthy (967, p. 33). The data are based o legth () ad timber volume () for 76 forest strips. Murthy (967, p.49) ad Kushwaha ad Sigh (989) reported the values of itraclass correlatio coefficiets ρ ad ρ approximately equal for the systematic sample of size 6 by eumeratig all possible systematic samples after arragig the data i ascedig order of strip legth. The details of populatio parameters are : N = 76, = 6, = 8.636, = , S = , S = , ρ = 0.870, 3 S = S 4 = Table shows the percetage relative efficiecy (PRE) of t (optimum) with respect to y for the differet choices of W ad L. Table : PRE of t (optimum) with respect to y W L PRE
8 Coclusio I this paper, we have proposed a geeral family of estimators of populatio mea i systematic samplig usig a auxiliary variable i the presece of orespose. The optimum property of the family has bee discussed. The study cocludes that the suggested family coverges to the usual regressio estimator of populatio mea i systematic samplig uder orespose if the parameter α attais its optimum value. From Table, it ca easily be see that the estimator t (optimum) performs always better tha the usual estimator y. It is also observed that the percetage relative efficiecy (PRE) of t (optimum) with respect to y decreases with icrease i orespose rate W as well as L. Refereces. Baarasi, Kushwaha, S.N.S. ad Kushwaha, K.S. (993): A class of ratio, product ad differece (RPD) estimators i systematic samplig, Microelectro. Reliab., 33, 4,
9 . Bucklad, W. R. (95): A review of the literature of systematic samplig, JRSS, (B), 3, Cochra, W. G. (946): Relative accuracy of systematic ad stratified radom samples for a certai class of populatio, AMS, 7, Fiey, D.J. (948): Radom ad systematic samplig i timber surveys, Forestry,, Hase, M. H. ad Hurwitz, W. N. (946) : The problem of orespose i sample surveys, Jour. of The Amer. Stat. Assoc., 4, Hasel, A. A. (94): Estimatio of volume i timber stads by strip samplig, AMS, 3, Khoshevisa, M., Sigh, R., Chauha, P., Sawa, N. ad Smaradache, F. (007): A geeral family of estimators for estimatig populatio mea usig kow value of some populatio parameter(s). Far East J. Theor. Statist.,, Kushwaha, K. S. ad Sigh, H.P. (989): Class of almost ubiased ratio ad product estimators i systematic samplig, Jour. Id. Soc. Ag. Statistics, 4,, Lahiri, D. B. (954): O the questio of bias of systematic samplig, Proceedigs of World Populatio Coferece, 6, Madow, W. G. ad Madow, L.H. (944): O the theory of systematic samplig, I. A. Math. Statist., 5, 4.. Murthy, M.N. (967): Samplig Theory ad Methods. Statistical Publishig Society, Calcutta.. Nair, K. R. ad Bhargava, R. P. (95): Statistical samplig i timber surveys i Idia, Forest Research Istitute, Dehradu, Idia forest leaflet, Queouille, M. H. (956): Notes o bias i estimatio, Biometrika, 43, Sigh, R ad Sigh, H. P. (998): Almost ubiased ratio ad product type estimators i systematic samplig, Questiio,,3, Sigh, R., Kumar, M. ad Smaradache, F. (008): Almost Ubiased Estimator for Estimatig Populatio Mea Usig Kow Value of Some Populatio Parameter(s). Pak. J. Stat. Oper. Res., 4() pp6376.
10 6. Sigh, R. ad Kumar, M. (0): A ote o trasformatios o auxiliary variable i survey samplig. MASA, 6:, Sukhatme, P. V., Paes, V. G. ad Sastry, K. V. R. (958): Samplig techiques for estimatig the catch of sea fish i Idia, Biometrics, 4, ates, F. (948): Systematic samplig, Philosophical Trasactios of Royal Society, (A), 4,
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