Estimation of Population Mean in Presence of Non-Response in Double Sampling

Transcription

1 J. Stat. Appl. Pro. 6, No. 2, (2017) 345 Joural of Statistics Applicatios & Probability A Iteratioal Joural Estimatio of Populatio Mea i Presece of No-Respose i Double Samplig G. N. Sigh, A. Kumar ad Gajedra K. Vishwakarma Departmet of Applied Mathematics, Idia Istitute of Techology Dhabad, Jharkhad , Idia Received: 9 Aug. 2016, Revised: 22 Apr. 2017, Accepted: 26 Apr Published olie: 1 Jul Abstract: The work doe i this article is cocered with the developmet ad efficiet estimatio procedure of populatio mea, usig iformatio o auxiliary variables at the estimatio stage i double samplig. We have proposed a geeral class of estimators ad the properties of the proposed estimator are examied. Theoretical coditios have bee made uder which our proposed class of estimators are superior to the existig oes. Empirical studies are carried out to assess the behaviours of the proposed class of estimators with respect to the exitig estimator. Keywords: Double samplig, study variable, auxiliary variable, bias, mea square error, o-respose 1 Itroductio It is well established fact that auxiliary iformatio i study of sample survey gives us a efficiet estimate of populatio parameters like as populatio mea or total, uder some crucial coditios. This iformatio may be used for drawig a radom sample usig simple radom samplig without replacemet (SRSWOR / SRSWR) simple radom samplig with replacemet, to stratificatio, systematic or probability proportioal to size samplig strategy or for estimatig the populatio parameter or at both purposes. Auxiliary iformatio gives us a variety of techiques by meas of ratio, product, regressio ad other methods. Icorporatig the kowledge of the auxiliary variables is very importat for the costructio of efficiet estimators for the estimatio of populatio parameters ad icreasig the efficiecy of the estimators i differet samplig desig. While coductig the sample surveys i the field of agriculture, social scieces ad medical scieces, the problem of o respose is very commo i practice. A estimate obtaied from such icomplete data may be misleadig especially whe the respodets differ from the o-respodets because the estimate ca be biased. The problem of estimatio of populatio mea usig the techique of sub samplig from o respodets was first itroduced by Hase ad Hurwitz [13]. It is well kow fact that i sample surveys precisio i estimatig the populatio mea may be icreased by usig iformatio o sigle or multiple auxiliary variables. Followig Hase ad Hurwitz [13] techique, several authors icludig Cochra [23], Rao [14,15], Khare ad Srivastava [5,6,7], Khare ad Rehma [1], Okafor ad Lee [8] ad Tabasum ad Kha [17,18] ad Sigh ad Kumar [10] have studied the problem of the estimatio procedure of populatio mea i presece of o-respose usig iformatio o auxiliary variable. Olki [12], Mohaty [19], Srivastava [21], Sigh ad Kumar [11], Khare ad Siha [4], Vishwakarma ad Sigh [9] ad others have made the extesio of the ratio method of estimatio to the case where multiple auxiliary variables are used to icrease the precisio of estimates. However, i may situatios of practical importace the problem of estimatio of populatio mea of the study variable y assumes importace whe the populatio mea of the auxiliary variable x is ot kow i presece of o-respose. I such a situatio the estimate of populatio mea of the x is furished from a large first phase sample of size draw from a populatio of uits by simple radom samplig without replacemet (SRSWOR). A smaller secod phase sample of size (i.e. < ) is draw from by SRSWOR ad the variable y uder ivestigatio is measured o it. This techique is kow as double samplig. Double samplig happes to be a powerful ad cost effective (ecoo- mical) techique for obtaiig the reliable estimate i first-phase (prelimiary) sample for the ukow Correspodig author amod.ism01@gmail.com Natural Scieces Publishig Cor.

2 346 G. N. Sigh et al.: Estimatio of populatio mea i presece of... populatio parameters of the auxiliary variables. For example, Okafor ad Lee [8] ad Tabasum ad Kha [17] have metioed that the procedure of double samplig ca be applied i a household survey where the household size is used as a auxiliary variable for the estimatio of family expediture. Iformatio ca be obtaied completely o the family size, while there may be o-respose o the household expediture. I the preset paper, we have proposed a geeral class of estimators for estimatig populatio mea usig auxiliary variable with double samplig i presece of o-respose. We have obtaied the expressios for bias ad mea square errors of the proposed class of estimators for the fixed value of ad, also for the optimum values of the costats. A illustratio of the proposed class of estimators has bee made with the relevat class of estimators. 2 Notatios The double samplig i presece of o-respose samplig scheme is that, let a fiite populatio U =(U 1,U 2,,U N ) of N uits y ad x are the variables uder study ad auxiliary variable respectively with populatio meas Ȳ ad X. Let y k ad x k be the values of y ad x for the k-th (k = 1,2,,N) uit i the populatio. If the iformatio o a auxiliary variable x whose populatio mea X is kow ad highly correlated to y is readily available for all the uits of the populatio, it is well kow that regressio ad ratio type estimators of populatio mea Ȳ could be used for good performace. However, i certai practical situatios whe populatio mea X is ot kow, a priori i such case the techique of two-phase samplig is useful. If there is o-respose i the secod phase sample oe may form a estimator by utilizig the iformatio oly from the respodets or take a sub-sample of the o-respodets ad re-cotact them. We assume that at the first phase sample of size, all the uits supplied iformatio o the auxiliary variables x ad at the secod phase sample of size, i which 1 uits supply iformatio ad 2 uits refuse to respod for study variable y ad as well as auxiliary x. Followig Hase ad Hurwitz [13] techique of sub-samplig the o-respodig group, a sub-sample of size m uits (m= 2 k,k>1) is selected at radom (without replacemet) from the 2 o- respodet uits, where k is the iverse samplig rate at the secod phase sample of size. All the m uits respod at this time ow ad the whole populatio (i.e. U) is supposed to be cosistig of two o-overlappig strata of N 1 ad N 2 uits. Stratum of N 1 respodig uits (deoted by U 1 ) would respod o the first call at the secod phase ad the stratum of N 2 (N 2 = N N 1 ) o-respodig uits (deoted by U 2 ) would ot respod o the first call at the secod phase but will respod o the secod call. Further, we assume that the strata sizes of N 1 ad N 2 are ot kow well i advace, see Tripathi ad Khare [22]. The stratum weights of respodig ad o-respodig groups are give by (W 1 = N 1 N ) ad (W 2 = N 2 N ) ad their estimates are cosidered as ( W 1 = N 1 N ) ad ( W 2 = N 2 N ) respectively. Let first ad secod phase sample be deoted by u ad u respectively ad let u 1 = u U 1 ad u 2 = u U 2. The sub-sample of u 2 will be deoted by u 2m. The followig are the list of otatios, cosidered for their further use: Ȳ = N i=1 y i N : The populatio mea of the study variable y. X = N i=1 x i : The populatio mea of the auxiliary variable x. Ȳ 1 = N 1 i=1 X 1 = N 1 i=1 N y i N 1 : The populatio mea of the study variable y 1 of the respose group. x i Ȳ 2 = N 1+N 2 i=n 1 +1 X 2 = N 1+N 2 i=n 1 +1 N 1 : The populatio mea of the auxiliary variable x 1 of the respose group. y i N 2 : The populatio mea of the study variable y 2 of the o-respose group. x i N 2 : The populatio mea of the auxiliary variable x 2 of the o-respose group. S 2 y = 1 (N 1) N i=1 (y i Ȳ) 2 : The populatio variace of the study variable y. S 2 x = 1 (N 1) N i=1 (x i X) 2 : The populatio variace of the auxiliary variable x. Sy 2 1 = 1 (N 1 1) N 1 i=1 (y i Ȳ 1 ) 2 : The populatio variace of the study variable y 1 of the respose group. Sx 2 1 = 1 (N 1 1) N 1 i=1 (x i X 1 ) 2 : The populatio variace of the auxiliary variable x 1 of the respose group. Sy 2 2 = 1 (N 2 1) N 1+N 2 i=n 1 +1 (y i Ȳ 2 ) 2 : The populatio variace of the study variable y 2 of the o-respose group. Sx 2 2 = 1 (N 2 1) N 1+N 2 i=n 1 +1 (x i X 2 ) 2 : The populatio variace of the auxiliary variable x 2 of the o-respose group. ρ yx = S yx S y S x : Correlatio coefficiet betwee the variable y ad x (i.e. U). The sample mea ȳ 1 = i=1 y i is ubiased for Ȳ 1, but has a bias equal to W 2 (Ȳ 1 Ȳ 2 ) i estimatig the populatio mea Ȳ. Natural Scieces Publishig Cor.

3 J. Stat. Appl. Pro. 6, No. 2, (2017) / The sample mea ȳ 2r = r i=1 y i r is ubiased for Y 2 of the 2 uits. Thus, a ubiased estimator for the populatio mea Ȳ is give by: ȳ = 1ȳ ȳ 2m ad x 2r = r i=1 x i r deoted the mea of sub-sample uits. A ubiased estimator for the populatio mea X is (1) x = 1 x x 2m (2) 3 Suggested class of estimators Motivated with the above work, usig ukow real costat W 1 ad W 2 ad oe auxiliary variables x, we defie a geeral class of estimators for estimatig Ȳ as follows. where ψ(w 1,w 2,h 2(0,1))=w 1 + w 2 h 2(0,1), 2 i=1 w i = 1 h 1 (α,β)=( x x ) α ( x x )β,h 2 (α,β)=( x x ) α ( x x )β,h 3 (α,β)=( x x ) α ( x x ) β p( x, x,ȳ )= x x ȳ T(w 1,w 2 )=ȳ h 3 ψ(w 1,w 2,h 2(0,1)) (3) Now we idetify some of the members of the proposed class of estimators preset below T 1 = ȳ h 2 (0,1)=ȳ x x (4) T 2 = ȳ h 2 (0, 1)= ȳ x x (5) T 3 = ȳ h 3 (1,0)=ȳ x x (6) T 4 = ȳ h 3 ( 1,0)=ȳ x x (7) T 5 = ȳ h 3 (1,0)h 2 (0,1)=ȳ x x x x (8) T 6 = ȳ (1+b p( x, x,ȳ ))=ȳ + b ( x x ) (9) where b = s ys is a estimator of populatio regressio coefficiet β = S yx based o secod phase ad s 2 x S 2 x s yx = 1 ( 1) ( u 1 x j y j + m u2m x j y j xȳ ),s 2 x = 1 ( 1) ( u 1 x 2 j + m u 2m x 2 j x x ) ad S yx = 1 (N 1) ( N i=1 (y i Ȳ)(x i X)) It may be oted that the estimators T i (i=1,2,,6) are well kow ratio, product, regressio ad chai type estimators i two phase samplig. The estimators (T 1,T 2,T 3,T 6 ) are first proposed ad studied by Khare ad Srivastava [5]. The estimator T 1 is revisited by Okafor ad Lee [8] ad Tabasum ad Kha [17]. Further, the estimator T 3 is recosidered by Tabasum ad kha [18] ad the estimator T 6 is revisited by Okafaor ad Lee [8]. The estimator T 5 is proposed by Sigh ad Kumar [10] i the presece of o-respose. Natural Scieces Publishig Cor.

4 348 G. N. Sigh et al.: Estimatio of populatio mea i presece of... 4 Some estimators of the proposed class It is also visible to ote that a umber of chai ratio type, regressio type ad other estimators fall uder the proposed class of estimators T(w 1,w 2 ) for differet choice of weights(w 1,w 2 ). (i)ratio type estimator T 1 proposed by Khare ad Srivastava [5], Okafor ad Lee [8] ad Tabasum ad Kha [17] (ii)chai-ratio type estimator T 5 proposed by Sigh ad Kumar [10] (iii)motivated by Chakrabarty [16] estimator T(1,0)=ȳ h 3 (1,0)ψ(1,0,h 2 (0,1)) (10) T(0,1)=ȳ h 3 (1,0)ψ(0,1,h 2 (0,1)) (11) T(1 α,α)=ȳ h 3 (1,0)ψ(1 α,α,h 2 (0,1)) (12) (iv)suggested by Ray et al. [20] T(1 α, α)= ȳ h 3 (1,0)ψ(1 α, α,h 2 (0,1)) (13) 5 Properties of estimator The bias ad mea square errors (MSE) of the proposed class of estimators T(w 1,w 2 ) to the first order of approximatios are derived uder large sample approximatios usig the followig trasformatios: ȳ = Ȳ(1+e 0 ), x = X(1+e 1 ), x= X(1+e 2 ), x = X(1+e 3 ) Such that e i <1(i=0,1,,3). Further, we have the followig expectatios. E(e i )=0,(i=0,1,,3),E(e 2 0 )= f 1Cy 2+W 2 Cy(2) 2,E(e2 1 )= f 1Cx 2+W 2 Cx(2) 2, E(e 2 2 )=E(e 1e 2 )= f 1 Cx 2,E(e2 3 )=E(e 1e 3 )=E(e 2 e 3 )= f 2 Cx 2,E(e 0e 2 )= f 1 ρ yx C y C x,e(e 0 e 3 )= f 2 ρ yx C y C x, E(e 0 e 1 )= f 1 ρ yx C y C x +W 2 ρ yx(2) C y(2) C x(2) where f 1 =( 1 N 1), f 2 =( 1 N 1), f 3 =( f 2 f 3 )=( 1 1 ) C y,c x : Coefficiet of variatios of the variables y ad x respectively based o the whole populatio. ρ yx(2) : Correlatio coefficiets betwee the variables show i suffice i the o-respose group of the populatio (i. e. U 2 ). Thus, expressig T(w 1,w 2 ) i terms of e s ad eglectig the terms of e s havig power greater tha two we get T(w 1,w 2)= Ȳ[R+1+W 1(e 0 e 1 + e 3 + e 2 1 e 0 e 1 + e 0 e 3 e 1 e 3 ) +W 2(e 0 e 1 e 2 + 2e 3 + e 2 1+ e 2 2+ e 2 3 e 0 e 1 e 0 e 2 + 2e 0 e 3 + e 1 e 2 2e 1 e 3 2e 2 e 3 )] (14) where R=(W 1 +W 2 1) Takig expectatios o both sides of the equatio (14) ad usig the expectatio values, we obtai the expressios for bias B(T) ad mea square errors M(T) of the class of estimators to the first order of approximatios as B(T)= Ȳ[R+W 1( f 3 Cx 2 f 3 ρ yx C y C x +W 2 (Cx(2) 2 ρ yx(2)c y(2) C x(2) )) +W 2 (3 f 3Cx 2 2 f 3ρ yx C y C x +W 2 (Cx(2) 2 ρ yx(2)c y(2) C x(2) ))] (15) ad where M(T)= Ȳ 2 [W 2 1 A+W 2 2 B+2W 1 W 2 C+2W 1 RD+2W 2RE] (16) Natural Scieces Publishig Cor.

5 J. Stat. Appl. Pro. 6, No. 2, (2017) / A=( f 1 C 2 y + f 3C 2 x 2 f 3ρ yx C y C x +W 2 (C 2 y(2) +C2 x(2) 2ρ yx(2)c y(2) C x(2) )) B=( f 1 C 2 y + 4 f 3 C 2 x 4 f 3 ρ yx C y C x +W 2 (C 2 y(2) +C2 x(2) 2ρ yx(2)c y(2) C x(2) )) C=( f 1 C 2 y + 2 f 3C 2 x 3 f 3ρ yx C y C x +W 2 (C 2 y(2) +C2 x(2) 2ρ yx(2)c y(2) C x(2) )) D=( f 3 C 2 x f 3ρ yx C y C x +W 2 (C 2 x(2) ρ yx(2)c y(2) C x(2) )) E =(3 f 3 C 2 x 2 f 3 ρ yx C y C x +W 2 (C 2 x(2) ρ yx(2)c y(2) C x(2) )) 6 Miimum MSE of suggested class of estimators Sice W 1 (i = 1,2) are ukow weights ad have a specific choice of these yields a particular member of the class T(w 1,w 2 ), it is desirable to detect that member of the class which has miimum MSE. This ca be achieved by miimizig MSE give i equatio (16) with respect to the ukow costat W 1 (i=1,2). Differetiatig equatio (16) with respect to W 1 ad W 2 ad equatig them zero, we have the optimum value of W 1 (i=1,2) as follows ad (W 1 ) opt = (1+D)(1+B+2E) (1+E)(1+C+D+E) = W (1+A+2D)(1+B+2E) (1+C+D+E) 2 1 (W 2 ) opt = (1+E)(1+A+2D) (1+D)(1+C+D+E) = W (1+A+2D)(1+B+2E) (1+C+D+E) 2 2 Substitutig these optimum values of W i (i = 1, 2) i equatio (16), we have miimum MSE of T as mi.m(t)= Ȳ 2 [W 2 1 A+W 2 2 B+2W 1 W 2 C+2W 1 RD+2W 2 RE] (17) 7 Efficiecies compariso I this sectio we ivestigate the situatios uder which our proposed class of estimators T(w 1,w 2 ) are preferable over the existig estimators such as Hase ad Hurwitz [13] sample mea estimator ȳ, T i (i = 1,2,,6). The variace V(.)/MSE of these estimators to the first order of approximatios are obtaied as V(ȳ )= Ȳ 2 [ f 1 Cy 2 +W 2 Cy(2) 2 ] (18) M(T 1 )= Ȳ 2 [ f 1 Cy 2 + f 3Cx 2 2 f 3ρ yx C y C c +W 2 Cy(2) 2 ] (19) M(T 2 )= Ȳ 2 [ f 1 Cy 2 + f 3 Cx f 3 ρ yx C y C c +W 2 Cy(2) 2 ] (20) M(T 3 )= Ȳ 2 [ f 1 Cy 2 + f 3Cx 2 2 f 3ρ yx C y C c +W 2 (Cy(2) 2 +C2 x(2) 2ρ yx(2)c y(2) C x(2) )] (21) M(T 4 )= Ȳ 2 [ f 1 Cy 2 + f 3Cx f 3ρ yx C y C c +W 2 (Cy(2) 2 +C2 x(2) 2ρ yx(2)c y(2) C x(2) )] (22) ad M(T 5 )= Ȳ 2 [ f 1 Cy f 3 Cx 2 4 f 3 ρ yx C y C c +W 2 (Cy(2) 2 +C2 x(2) 2ρ yx(2)c y(2) C x(2) )] (23) M(T 6 )= Ȳ 2 [ f 1 Cy 2 f 3 ρyxc 2 x]+w 2 2 [Sy(2) 2 + β yxsx(2) 2 (β yx 2β yx(2) )] (24) It may be oted that the expressio of the MSE of T show i equatio (17) is quite difficult. However, the performace of the proposed class of estimators is examied through empirical study over differet populatio which established the superiority over the traditioal oes. Natural Scieces Publishig Cor.

6 350 G. N. Sigh et al.: Estimatio of populatio mea i presece of... 8 Empirical study To see the performace of the proposed class of estimators of the populatio mea, we cosider three atural dataset of the variables y ad x ad the values of the various parameters are give as follows. Populatio I- Source: [Khare ad Siha [2]] The data belogs to the data o physical growth of upper-socio-ecoomic group of 95 school childre of Varaasi uder a ICMR study, Departmet of Pediatrics, BHU durig has bee take uder study. The first %25 (i.e. 24 childre) uits have bee cosidered as o-respose uits. The values of the parameters related to the study variable y (the weight i kg) ad the auxiliary variable x (the skull circumferece i cm) have bee give below. It is to be oted that this populatio was also cosidered by several authors icludig Sigh ad Kumar [11]. N = 25,Ȳ = 19.49, X = 51.17,C y = 0.15,C X = 0.03,C y(2) = 0.12,C x(2) = 0.02, ρ yx = 0.32,ρ yx(2) = 0.47,N 2 = 24,W 2 = Populatio II- Source: [District Cecus Hadbook, 1981, Orissa, Published by Govt. of Idia] The 109 Village / Tow / Ward wise populatio of urba area uder Police-statio-Baria, Tahasil-Champua, Orissa, Idia has bee take uder study. The last %25 villages (i. e. 27 villages) have bee cosidered as o-respose group of the populatio. The study variable (y) is umber of literate persos i the village while the umber of mai workers i the village is cosidered as auxiliary variable (x). This populatio was also cosidered as umerical evidece i the works of several authors icludig Khare ad Siha [4]. N = 109,Ȳ = , X = ,C y = 0.76,C X = 0.68,C y(2) = 0.68,C x(2) = 0.057, ρ yx = 0.81,ρ yx(2) = 0.78,N 2 = 28,W 2 = Populatio III- Source: [District Cecus Hadbook, 1981, West Begal, Published by Govt. of Idia] Niety-six village wise populatio of rural area uder Police-statio-Sigur, District-Hooghly, West Begal has bee take uder the study. The %25 villages (i.e. 24 villages) whose area is greater tha 160 hectares have bee cosidered as o-respose group of the populatio. The umber of agricultural labours i the village is take as study variable (y) while the area (i hectares) of the village is take as auxiliary variables (x). It is to be oted that this populatio was also cosidered by Khare ad Siha [3]. N = 96,Ȳ = , X = ,C y = 1.32,C X = 0.81,C y(2) = 2.08,C x(2) = 0.094, ρ yx = 0.77,ρ yx(2) = 0.72,N 2 = 24,W 2 = Here we have computed the percetage relative efficiecy (PRE) of propose estimator ad other exitig Hase ad Hurwitz [13] sample mea estimatorȳ,t 1,T 3 ad T 5 with respect to usual ubiased estimator ȳ. PRE = V(ȳ ) M(ι) 100whereι =(ȳ,t 1,T 3,T 5,T) (25) 9 Iterpretatios of results The followig iterpretatios may be read out from Table 1 (i)for all populatio I, II ad III, the PRE of estimator T 1 decreasig for differet choice of the sub-samplig fractio ( 1 k ) ad sample size (, ). (ii)for icrease value of the sub-samplig fractio ( 1 k ) ad sample size (, ), the PRE for populatio I, II ad III of estimators T 3,T 5 ad T are icreasig. (iii)it is also visible that the estimator T 5 i populatio-ii do ot gai efficiecy for the value k=2 at = Coclusios From above aalyses, it is clear that the proposed class of estimators T cotribute sigificatly to hadle the differet realistic situatios of o-resposes while estimatig populatio mea i double samplig. It is visible that the proposed class of estimators is more efficiet tha the other existig estimators uder the similar realistic situatios. Natural Scieces Publishig Cor.

7 J. Stat. Appl. Pro. 6, No. 2, (2017) / Table 1: PRE of the differet estimators with respect to ȳ Populatio I ȳ T 1 T 3 T 5 T ȳ T 1 T 3 T 5 T k =65 = Populatio II ȳ T 1 T 3 T 5 T ȳ T 1 T 3 T 5 T k =75 = * * Populatio III ȳ T 1 T 3 T 5 T ȳ T 1 T 3 T 5 T k =50 = Natural Scieces Publishig Cor.

8 352 G. N. Sigh et al.: Estimatio of populatio mea i presece of... Ackowledgemet The authors are very much thakful to the Idia Istitute of Techology(ISM), Dhabad. Authors are also thakful to the aoymous referee for his valuable suggestios that improved this paper. Refereces [1] B.B. Khare ad H.U. Rehma, Iteratioal Joural Statistics Ecoomics 15, (2014). [2] B.B. Khare ad R.R. Siha, Samplig Theory ad Quality Cotrol 1, (2007). [3] B.B. Khare ad R.R. Siha, Statistics i Trasitio-ew series 10, 3-14 (2009). [4] B.B. Khare ad R.R. Siha, Statistica 49, (2012). [5] B.B. Khare ad S.R. Srivastava, Natioal Academy of Sciece Letters Idia 16, (1993). [6] B.B. Khare ad S.R. Srivastava, Proceedigs of Natioal Academy of Sciece Idia 65, (1995). [7] B.B. Khare ad S.R. Srivastava, Commuicatio i Statistics- Theory Methods 26, (1997). [8] F.C. Okafor ad H. Lee, Survey Methodology 26, (2000). [9] G.K. Vishwakarma ad R. Sigh, Joural of Computatios ad Mdellig 6, (2016). [10] H.P. Sigh ad S. Kumar, Statistical Papers 51, (2010 a). [11] H.P. Sigh ad S. Kumar, Joural of Statistical Plaig ad Iferece 140, (2010 b). [12] I. Olki, Biometrika 45, (1958). [13] M.H. Hase ad W.N. Hurwitz, Joural of America Statistical Associatio 41, (1946). [14] P.S.R.S. Rao, Survey Methodology 12, (1986). [15] P.S.R.S. Rao, Paper preseted at a special cotributed sessio of the Iteratioal Statistical Associatio Meetig Sept Tokyo, Japa (1987). [16] R.P. Chakrabarty, Cotributio to the theory of ratio-type estimators, Upublished Ph.D. Thesis, Texas A ad M Uiversity, College Statio, U. S. A, (1968). [17] R. Tabasum ad I.A. Kha, Joural of the Idia Society of Agricultural Statistics 58, (2004). [18] R. Tabasum ad I.A. Kha, Assam Statistical Review 20, (2006). [19] S. Mohaty, Joural of Idia Statistical Associatio 5, 1-14 (1967). [20] S.K. Ray, A. Sahat ad A. Sahai, Aals of the Istitute of Mathematical Statistics 31, (1979). [21] S.K. Srivastava, Joural of America Statistical Associatio 66, (1971). [22] T.P. Tripathi ad B.B. Khare, Commuicatios i Statistics - Theory ad Methods 26, (1997). [23] W.G. Cochra, Samplig Techiques (3rd ed.), Joh Wiley ad Sos, New York, (1977). G. N. Sigh is a Professor of Statistics i the Departmet of Applied Mathematics, Idia Istitute of Techology (ISM) Dhabad, Idia. He obtaied his Ph.D. degree i 1990 from Baaras Hidu Uiversity, Varaasi, Idia. He has more tha 27 years of teachig experiece i the field of statistics. He served as faculty i Pajab Uiversity, Chadigarh ad Kurukshetra Uiversity, Idia. He has more tha 30 years of research experiece i the various field of Statistic which covers Sample Surveys, Statistical Iferece, Data Aalysis, Data Miig etc. He has published umber of research papers i Idia ad Foreig jourals of repute. He preseted his research problems i iteratioal ad atioal cofereces ad delivered various ivited talks i academic forum. He has produced 12 Ph.D, 5 M.phil ad 4 research projects. Amod Kumar is a research scholar i the Departmet of Applied Mathematics, Idia Istitute of Techology (ISM) Dhabad, Idia. He is pursig Ph.D. i Applied Statistics. His research iterest is i the areas of Sample Survey ad Statistical Iferece. Natural Scieces Publishig Cor.

9 J. Stat. Appl. Pro. 6, No. 2, (2017) / Gajedra Kumar Vishwakarma is a Assistat Professor of Statistics i the Depart of Applied Mathematics, Idia Istitute of Techology (ISM) Dhabad, Idia. Gajedra Kumar Vishwakarma is a Assistat Professor of Statistics i the Departmet of Applied Mathematics, Idia Istitute of Techology (ISM) Dhabad, Idia. He obtaied his Ph.D. degree i 2007 from Vikram Uiversity, Ujjai, Idia. He has several years of academic as well as idustrial research experiece i the field of applied statistics. His research experiece covers both applied as well as theoretical provices. He served as visitig scietist cum faculty i Idia Statistical Istitute, North-East Cetre, Tezpur (Assam), Idia ad as a Associate Scietist i Lupi Research Park, Pue, Idia. He is elected Fellow of the Society of Earth Scietists, Idia, a Elected Member of Iteratioal Statistical Istitute, Netherlads ad member Fellow of Royal Statistical Society, UK. He is member of the advisory boards ad editorial board member of several jourals. He has published umber of research papers i iteratioal joural reputes. He preseted his research problems i iteratioal ad atioal cofereces ad delivered various ivited talks at idustrial as well as academic forum. He received Youg Scietist Award from Ceter for Advaced Research ad Desig, Cheai, Idia. Natural Scieces Publishig Cor.