A GENERAL CLASS OF ESTIMATORS IN TWO-STAGE SAMPLING WITH TWO AUXILIARY VARIABLES

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1 Hacettepe Journal of Mathematics and Statistics Volume 40(5) (011) A GENEAL CLASS OF ESTIMATOS IN TWO-STAGE SAMPLING WITH TWO AUXILIAY VAIABLES L.N. Sahoo.K. Sahoo S.C. Senapati and A.K. Mangaraj eceived 08:03:009 : Accepted 10:03:011 Abstract This paper presents a general class of estimators for a finite population total with the aid of two auxiliary variables in a two-stage sampling with varying probabilities. The methodology developed can be extended readily to three-stage and stratified two-stage sampling designs. Keywords: Asymptotic variance Auxiliary variable Two-stage sampling. 000 AMS Classification: 6 D Introduction Consider U a finite population consisting of N first stage units (fsu) U 1U...U N such that U i contains M i second stage units (ssu) and M = N Mi. Let Yi Xi and Z i be the totals of U i in respect of the study variable y and two auxiliary variables x and z respectively with corresponding overall totals Y = N Yi X = N Xi and Z = N Zi. To estimate Y let us consider a general class of two-stage sampling designs: At stage one a sample s (s U) of n fsus is drawn from U according to any design with π i and π ij as the known first and second order inclusion probabilities. Then for every i s a sample s i of m i ssus is drawn from U i (s i U i) with suitable selection probabilities at the second stage. More detailed accounts of two-stage sampling procedure are given in general survey sampling books (cf. Cochran [1] Sarndal et al [9]). Let E 1E (V 1V ;Cov 1Cov ) denote the expectation (variance covariance) operators over repeated sampling in the first and second stages; by E (V or Cov) we denote the overall expectation (variance or covariance). It is assumed that from the second Department of Statistics Utkal University Bhubaneswar India. (L. N. Sahoo. K. Sahoo) lnsahoostatuu@rediffmail.com Corresponding Author. Department of Statistics avenshaw College Cuttack India. scsenapati00@rediffmail.com Department of Statistics.D. Women s College Bhubaneswar India. akmangaraj@gmail.com

2 758 L.N. Sahoo.K. Sahoo S.C. Senapati A.K. Mangaraj stage sample s i i s unbiased estimates t ix and t iz respectively for Y ix i and Z i are available. Then V () = σ iy V (t ix) = σ ix V(t iz) = σ iz Cov (t ix) = σ iyx Cov (t iz) = σ iyz Cov (t ixt iz) = σ ixz. Given the first stage sample s we define π-estimators t y = π i t x = and t z = t iz π i so that E(t y) = Y E(t x) = X E(t z) = Z V(t y) = σy + N V(t x) = σx + N σix π i V(t z) = σz + N σiz π i Cov(t yt x) = σ yx + N Cov(t yt z) = σ yz + N and Cov(t xt z) = σ xz + N σ ixz π i where σ y = 1 σ yx = 1 i j=1 σ iyz π i Yi (π iπ j π ij) Yj π i π j Yi (π iπ j π ij) Yj Xi Xj etc. π i π j π i π j i j=1 t ix π i σiy π i σ iyx π i We have seen that many two-stage sampling estimation techniques require advance knowledge on overall population totals X and Z. However these techniques do not fully exhaust the information content of the auxiliary variables. In many surveys when the clusters are selected it is more likely that the totals X i and Z i are known (or can be found easily or cheaply) for i s i.e. for the selected clusters. For instance in a crop survey if yx and z are respectively yield of the crop area under the crop and area under cultivation then information on total area under the crop (X i) and total area under cultivation (Z i) for the ith selected block (cluster of villages) can be obtained easily from the block records. Detail studies on the profitability of using knowledge on the auxiliary variable values at the level of first stage units are provided by several authors see for example Sahoo [4] Sahoo and Panda ([56]) Sahoo and Sahoo [7] Sahoo et al [8] Hansen et al [] and Smith [11]. In this context we may also refer to Zheng and Little [14] who used penalized spline nonparametric regression models on the selection of probabilities of the first stage units and Kim et al [3] who considered nonparametric regression estimation of the population total in which complete auxiliary information is available for the first stage units. Works of Singh et al [10] and Tracy and Singh [13] also focus on the use of various kinds of auxiliary information at different phases of a survey operation under a two-phase sampling set-up. In this paper we are aimed at constructing a general class of estimators for Y with explicit involvement of x and z motivated by the assumption that X Z X i and Z i i s are known.. The class of estimators For given s i following the work of Srivastava [1] let us define a class of estimators for Y i byŷi = gi(tiytixtiz) i s wheregi(tiytixtiz) isaknownfunctionof tiytix andtiz which may depend on X i and Z i but is independent of Y i such that g i(x iz i) = which implies that g i(y ix iz i) = Y i. Also for given s let t y = Ŷ i π i t x = X i π i t z = Z i π i and g(t yt xt z) be a function of t y t x and t z which may depend on X and Z butis independentofy suchthatg(t yxz) = t y whichimplies thatg(yxz) = Y. Further following Srivastava [1] let us consider the following assumptions: (a) (t ixt iz) i s and (t yt xt z) assume values in a bounded convex subspace 3 of 3-dimensional real space containing the points (Y ix iz i) and (YXZ) and

3 Estimators in Two-Stage Sampling 759 (b) The functions g i(t ixt iz) and g(t yt xt z) are continuous having first and second order partial derivatives w.r.t. their arguments which are also continuous in 3. Then motivated by the work of Srivastava [1] we propose a class of estimators of Y defined by t g = g(t yt xt z). Here Ŷ i = g i(t ixt iz) covers both linear and nonlinear functions of the statistics t ix and t iz. So it is impossible to obtain an exact general expression for the conditional variance V (Ŷi) due to the fact that the expectation operator is a linear operator. However for simplicity we use Taylor linearization technique (cf. Sarndal et al [9]) to approximate Ŷi by a more easily handled linear function so that an approximate expression for V (Ŷi) can be obtained. Hence on considering the first order Taylor approximation of thefunctiong i after expandingaround thepoint(y ix iz i)andneglecting theremainder term we obtain (1) Ŷ i Y i +g i0( Y i)+g i1(t ix X i)+g i(t iz Z i) where g i0g i1 and g i are respectively the first order differential coefficients of g i w.r.t. t ix and t iz when evaluated at (Y ix iz i). Hence from (1) and noting that g i0 = 1 to a first order of approximation we have E (Ŷi) Yi and () V (Ŷi) σ iy +g i1σ ix +g iσ iz +g i1σ iyx +g iσ iyz +g i1g iσ ixz. In light of the above discussion once again using the linear approximation (3) t g Y +(t y Y)+g 1(t x X)+g (t z Z) the asymptotic variance of t g is obtained as (4) V(t g) V(t y)+g 1V(t x)+g V(t z)+g 1Cov(t yt x)+g Cov(t yt z) +g 1g Cov(t xt z) whereg 1 andg are respectivelythe firstderivativesofg w.r.t. t x andt z around (YXZ). Verifying that V(t y) = V 1E (t y)+e 1V (t y) on simplification we get (5) V(t y) = σ y + V /π (Ŷi) i. Similarly under the conditional argument we also have Cov(t yt x) = σ yx Cov(t yt z) = σ yz Cov(t xt z) = σ xz V(t x) = σ x and V(t z) = σ z. Finally the formula for the asymptotic variance of t g is obtained as (6) V(t g) = σy +g1σ x +gσ z +g 1σ yx +g σ yz +g 1g σ xz / + σ iy +gi1σ ix +giσ iz +g i1σ iyx +g iσ iyz +g i1g iσ ixz πi. This variance is minimized when βiyx βiyzβizx βiyz βiyxβixz g i1 = = ĝ i1 (say) g i = = ĝ i (say) 1 β izxβ ixz 1 β izxβ ixz βyx βyzβzx βyz βyxβxz g 1 = = ĝ 1 (say) and g = = ĝ (say) 1 β zxβ xz 1 β zxβ xz where β iyx = σ iyx / σ ix β yx = σ yx / σ x etc. The optimum values of g i1g ig 1 and g determined are unique in the sense that they do not depend on each other for their computation. Using these optimum values

4 760 L.N. Sahoo.K. Sahoo S.C. Senapati A.K. Mangaraj in (6) we obtain a minimum asymptotic variance (which may be called the asymptotic minimum variance bound of the class) as (7) minv(t g) = σ y(1 ρ )+ where ρ i = ρ iyx +ρ iyz ρ iyxρ iyz ρ ixz 1 ρ ixz σiy(1 ρ i) / π i and ρ = ρ yx +ρ yz ρyxρyzρxz 1 ρ xz are such that ρ iyx = σ iyx/σ iyσ ix ρ yx = σ yx/σ yσ x etc. The estimator attaining this bound (which may be called a minimum variance bound (MVB) estimator) is a regression-type estimator defined by t = [ ĝ i1(t ix X i) ĝ i(t iz Z i)] ĝ 1(t x X) ĝ (t z Z). 3. Some specific cases of the class If there is no use for x and z Ŷi = tiy = tg = ty the simple expansion estimator of Y. On the other hand if the emphasis is laid on the use of either x or z or both t g defines a wide class of estimators. For various choices of g i and g it also reduces to many other classes. Let us now examine a few specific cases When the values of X and Z are not taken into consideration t g = t y producing a family of separate variety of estimators whose asymptotic variance structure is shown in (5). The minimum variance bound and the corresponding MVB estimator of the class are given by minv(t g) = σy + σiy(1 ρ i) / π i (8) t (s) = [ ĝ i1(t ix X i) ĝ i(t iz Z i)]. 3.. When X is unknown but X iz i for i s and Z are known we have Ŷi = g i(t ixt iz) and t g = g(t yt z). Then the asymptotic MVB of the class is given by (9) minv(t g) = σ y(1 ρ yz)+ and the MVB estimator is defined by σiy(1 ρ i) / π i t (1) = [ ĝ i1(t ix X i) ĝ i(t iz Z i)] β yz(t z Z) Assuming that X i(i s) Z are known and X Z i(i s) are unknown and then defining Ŷi = gi(tiytix) and tg = g(t yt z) we see that the class of estimators considered by Sahoo and Panda [6] is a particular case of t g. Here the MVB of the class and the resulting MVB estimator are given by (10) minv(t g) = σ y γ {σ z + } / σiz 1 ρ ixz πi + t () = [ γ i(t ix X i)] γ(t z Z) / σiy 1 ρ iyx πi where γ i = (β iyx +γβ izx) and γ = σyz+ N σ iz(β iyz β iyx β ixz) σ z + N σ iz(1 ρ ixz).

5 Estimators in Two-Stage Sampling AssumethatZ is unknownbutx iz i for i s andx are known. Then considering Ŷ i = g i(t iz) the class of estimators is defined by t g = g(t yt x) as studied by Sahoo and Sahoo [7]. In this case the minimum variance bound is (11) minv(t g) = σ y(1 ρ yx)+ σiy(1 ρ iyz) / π i and the corresponding MVB estimator is t (3) = [ β iyz(t iz Z i)] β yx(t x X) If the estimation procedure is carried out with the involvement of x only then Ŷ i = g i(t ix) and t g = g(t yt x) a class of estimators considered by Sahoo and Panda [5]. The asymptotic MVB of the class is (1) minv(t g) = σ y(1 ρ yx)+ σiy(1 ρ iyx) / π i and the corresponding MVB estimator is of the form t (4) = [ β iyx(t ix X i)] β yx(t x X) which can be transformed to the regression-type estimator developed by Sahoo [4] on adopting the simple random sampling without replacement (SSWO) design at different stages If s = U then π i = π ij = 1 for all i and j. In this case t g simply defines a class of estimators for a stratified sampling with N fsus as a set of strata If s i = U i i t g defines a class of estimators for single-stage cluster sampling. 4. Precision of the class In order to study precision of t g compared to other classes of estimators utilizing information on two auxiliary variables let us now consider the classes of estimators developed by Srivastava [1] and Sahoo et al [8]. These classes are respectively defined by and t c = h(t yt xt z) t l = f(ỹ X) where Ỹ = φi(tiytix)/πi and X = φ(t xt z) are such that the functions involved in composing the classes admit regularity conditions. It may be remarked here that t l makes use of pre-assigned values of XZX i and Z i (i s) whereas t c makes use of only X and Z.

6 76 L.N. Sahoo.K. Sahoo S.C. Senapati A.K. Mangaraj (13) (14) The asymptotic expressions for V(t c) and V(t l ) are given by V(t c) = σy +h 1σx +h σz +h 1σ yx +h σ yz +h 1h σ xz [ ]/ + σ iy +h 1σix +h σiz +h 1σ iyx +h σ iyz +h 1h σ ixz πi V(t l ) = σy +f1 σ x +φ σz +φ σ xz +f1(σ yx +φ σ yz) ( + σ iy +φ i1σix )/ +φ i1σ iyx πi where h 1 = h(tytxtz) (YXZ) t x h = h(tytxtz) (YXZ) t z φ i1 = φ i( t ix ) t ix φ = (Yi X i ) and f 1 =. (XZ) (YX) φ(t x t z ) t z f(ỹ X) X The MVB and the corresponding MVB estimators of t c and t l are (15) minv(t c) = σy(1 )+ σiy(1 ) / π i (16) minv(t l ) = σ y(1 ρ )+ σiy(1 ρ iyx) / π i t (c) = ty β1(tx X) β(tz Z) t (l) = [ β iyx(t ix X i)] ˆf 1(t x X) ˆφ (t z Z) whereis themultiple correlation coefficient oft y ont x andt z; β 1 andβ are respectively the partial regression coefficients of t y on t x and t y on t z; ˆf 1 = ĝ 1 ˆφ = βyz βyxβxz β yx β yzβ xz. As t c and t l can be taken to be potential competitors of t g one should naturally be interested to compare their precisions. But comparing (6) with (13) and (14) we can derive only some sufficient conditions under which an estimator of t g is asymptotically more precise than an estimator of t c or t l. However these conditions are extremely complicated and mainly depend on the choices of the functions h fφφ ig and g i and cannot lead to any straightforward conclusion unless the nature of these functions are known. But for simplicity if we accept MVB as an intrinsic measure of the precision of a class the problem of precision comparison seems to be easier and our attention will be concentrated on the MVB estimators only. Thus minv(t g) minv(t c) i.e. t is more precise than t (c) if ρ and ρi i and minv(t g) minv(t l ) i.e. t is always more precise than t (l). On these grounds we also find that t is more precise than t (s) t(1) t(3) whereas no conclusion can be drawn regarding the precision of t over t () 5. A simulation study. and t(4) As seen above a theoretical comparison is not very useful in showing the merits of the suggested estimation procedure over others. Therefore as a counterpart to the theoretical comparison we carry out a simulation study. In this study we do not limit ourselves to the MVB estimators only. The simulation study reported here involves repeated draws of independent samples from a natural population consisting of 198 blocks (ssus) divided into N = 7 wards (fsus) of Berhampur City of Orissa (India). The number of blocks (M i) in the 7 wards

7 Estimators in Two-Stage Sampling 763 are and 6. Three variables viz. number of educated females number of households and the female population are used as yx and z respectively data on which is readily available from the Census of India (1971) document. The estimators under consideration are the eight MVB estimators viz. t (s) t(1) t() t (3) t(4) t(c) t(l) and t and their respective ratio counterparts defined by t (s) = / X i Z i π i t (1) t ix t = X i Z i iz t ix t iz t () = X i Z t (3) t ix t = Z i z t iz t (4) = X i t ix t (l) = X i X t ix t x X t (c) t = X ty x X t x Z t z Z t x t z Z X i Z i X and t t = z t ix t iz t x We did not touch the product or product-type estimators as y is positively correlated with x and z. It may also be noted here that for simplicity we compute t and the other MVB estimators by considering population values of their respective coefficients and in this case the estimators are unbiased. The following performance measures of an estimator t are taken into consideration: (1) elative absolute bias (AB) = 100 B(t) /Y where B(t) = E(t) Y is the bias of t. () Percentage relative efficiency (PE) compared to the direct estimator t y = π i i.e. PE = 100V(t y)/v(t) where V(t) is the variance of t. Our simulation consisted in the selection of independent first stage samples each of size n = 10 fsus from the population by SSWO. From every selected U i (i = ) in a first stage sample a second stage sample of size m i = or 3 ssus is again selected by SSWO. Thus we now have independent samples each of size 0 or 30 ssus. Considering these independent samples simulated biases and variances of the comparable estimators are calculated. If r indexes the r-th sample the simulated bias and variance of an estimator t are given by B(t) = 1 t (r) Y r=1 Z. t z and V(t) = 1 ( r=1 t (r) 1 ) t (r) r=1 respectively where t (r) is the value of t for the r-th realized sample. The simulated values of AB and PE of different estimators are then calculated as suggested above and their values are displayed in Table 1. But we see that the simulated values of AB for the MVB estimators are not equal to zero as these values are computed from a limited number of independent samples.

8 764 L.N. Sahoo.K. Sahoo S.C. Senapati A.K. Mangaraj Table 1. AB and PE of Different Estimators Estimator B PE m i = m i = 3 m i = m i = 3 t (s) t (1) t () t (3) t (4) t (c) t (l) t t (s) t (1) t () t (3) t (4) t (c) t (l) t Simulation results show that t is superior to the other ratio-type estimators on the grounds of AB and PE. On the other hand t is superior to the other regressiontype (MVB) estimators with respect to AB for m i = 3 and PE. But it is slightly inferior to t (l) with respect to AB for mi =. This imperfection is probably caused by our restriction to independent samples. An increase in the number of samples and their size may improve the degree of performance of t considerably. However our simulation study though of limited scope clearly indicates that there are practical situations which can favor the application of the suggested estimation methodology. Acknowledgement The authors are grateful to the referees whose constructive comments led to an improvement in the paper. eferences [1] Cochran W. G. Sampling Techniques 3rd edition (John Wiley & Sons New York 1977). [] Hansen M.H. Hurwitz W.N. and Madow W.G. Sampling Survey Methods and Theory vol.i (John Wiley & Sons New York 1953). [3] Kim J.-Y. Breidt F. J. and Opsomer J. D. Nonparametric regression estimation of finite population totals under two-stage sampling (Technical eport 4 Department of Statistics Colorado State University 009). [4] Sahoo L. N. A regression-type estimator in two-stage sampling Calcutta Statistical Association Bulletin

9 Estimators in Two-Stage Sampling 765 [5] Sahoo L.N. and Panda P. A class of estimators in two-stage sampling with varying probabilities South African Statistical Journal [6] Sahoo L. N. and Panda P. A class of estimators using auxiliary information in two-stage sampling Australian & New Zealand Journal of Statistics [7] Sahoo L.N. and Sahoo.K. On the construction of a class of estimators for two-stage sampling Journal of Applied Statistical Science [8] Sahoo L. N. Senapati S. C. and Singh G. N. An alternative class of estimators in twostage sampling with two auxiliary variables Journal of the Indian Statistical Association [9] Sarndal C. E. Swensson B. and Wretman J. Model Assisted Survey Sampling (Springer- Verlag New York 199). [10] Singh V. K. Singh Hari P. and Singh Housila P. Estimation of ratio and product of two finite population means in two-phase sampling Journal of Statistical Planning and Inferences [11] Smith T. M. F. A note on ratio estimates in multi-stage sampling Journal of the oyal Statistical Society A [1] Srivastava S. K. A class of estimators using auxiliary information in sample surveys Canadian Journal of Statistics [13] Tracy D.S. and Singh H.P. A general class of chain regression estimators in two-phase sampling Journal of Applied Statistical Science [14] Zheng H. and Little. J. Penalized spline nonparametric mixed models for inference about finite population mean from two-stage sampling Survey Methodology