Ratio-cum-product and dual to ratio-cum-product estimators

Transcription

1 Ratio-cum-product and dual to ratio-cum-product estimators Ignas Bartkus 1 1 Vilnius Pedagogical University, Lituania ignas.bartkus@gmail.com Abstract Ratio-cum-product and dual to ratio-cum-product estimators are compared by simulation for stratified sampling design. Te definition of dual variable presented in Plikusas (2008) is used to define te dual variable for stratified sample design. 1 Introduction Te product ratio estimators using one dual auxiliary variable were introduced by Bandyopadyay (1980) and Srivenkataramana (1980). Ratio type estimators tat uses two auxiliary variables are also considered in te literature. Tese estimators are defined for simple random sampling and can be effective wen one auxiliary variable is positively correlated and te oter variable is negatively correlated wit te study variable. Te ratio-cum-product estimator is presented in Sing (1969), and dual to ratio-cum-product estimator in Sing et all (2005). In te paper Sing et all (2005) te dual to ratio-cum-product estimator is considered for simple random sample witout replacement. Te ratio estimators expressed as a weigted sum of some ratio and direct estimator are analyzed in Sing (2000). 2 Ratio-cum-product estimator 2.1 Simple random sample case Consider a finite population U = (u 1, u 2,..., u N ) of N units. Let a sample s of size n be drawn from tis population by simple random sampling witout replacements. Let y k represents te value of a response variable y and two auxiliary variables x and z are available. Te ratio and product estimators are ˆt y = N n k s ˆt x = N x k, n k s ˆt R = ˆt y ˆt x t x, Te ratio-cum-product estimator is defined as ˆt P = ˆt y ˆ, ˆ = N z k, t x = n k s ˆt RP = ˆt y ˆ ˆt x t x. x k, = z k.

2 Te approximate variances of te estimators ˆt R, ˆt P, ˆt RP are MSE(ˆt R) = N 2 1 f n Ȳ 2( C 2 y + C 2 x(1 2K yx) ), MSE(ˆt P ) = N 2 1 f n Ȳ 2( C 2 y + C 2 z (1 + 2K yz) ), MSE(ˆt RP ) = N 2 1 f n Ȳ 2( C 2 y + C 2 z (1 + 2K yz) + C 2 x(1 2K yx 2K zx) ), f = n, Kyx = ρyxcy/cx, Kzx = ρzxcz/cx, Kyz = ρyzcy/cz, N C y = sȳ Y, s2 y = 1 N 1 (y k Ȳ )2, Ȳ = 1 N ρ yx = syx s ys x, s yx = 1 N 1 (y k Ȳ )(x k X), C x, C z, ρ yz, ρ zx, X, Z are defined analogously and respective to te subscripts used. 2.2 Stratified simple random sample case Assume te population U consists of H strata: U = U 1... U H. Te size of stratum U is, and te size of simple random sample s in stratum U is, = 1,..., H. Te ratio and product estimators are ˆt yst = ˆt Rst = ˆt yxt ˆt xst t x, ˆt xst = Te ratio-cum-product estimator is defined as ˆt P st = ˆt yst ˆst, ˆt RP st = ˆt yst ˆst ˆt xst t x. And te approximate variances of tese estimators are MSE(ˆt Rst) = x k, ˆst = Ȳ 2( Cy 2 + Cx(1 2 2K yx ) ), z k, MSE(ˆt RP st) = MSE(ˆt P st) = Ȳ 2( Cy 2 + Cz( K zx ) ), Ȳ 2( Cy 2 + Cz( K yz ) + Cx(1 2 2K yx 2K zx ) ), f =, K yx = ρ yx C y /C x, K zx = ρ zx C z /C x, K yz = ρ yz C y /C z, C y = s y Ȳ, s2 y = 1 1 (y k Ȳ) 2, Ȳ = 1 k U k U Ȳ = 1 N

3 ρ xy = s xy 1, s xy = (y k s x s y 1 Ȳ)(x k X ), k U Estimators of te approximate variances are MSE(ˆt RP st) = MSE(ˆt Rst) = MSE(ˆt P st) = Ȳ 2( Ĉy 2 + n Ĉ2 x(1 2 ˆK yx ) ), Ȳ 2( Ĉy 2 + n Ĉ2 z(1 + 2 ˆK zx ) ), Ȳ 2( Ĉy 2 + n Ĉ2 z(1 + 2 ˆK yz ) + Ĉ2 x(1 2 ˆK yx 2 ˆK zx ) ), 3 Dual to ratio-cum-product estimator 3.1 Simple random sample case Te dual variable for te improvement of ratio-type estimator was used by Bandyopadyay (1980) and Srivenkataramana (1980). An estimator wit two auxiliary variables were considered by Sing et al (2005). Consider again simple random sampling of size n and introduce te linear transformation of te variables x and z: g = n/(n n). Ten x k = (1 + g) X gx k, z k = (1 + g) Z gz k, for k = 1,..., N, x = (1 + g) X g x, z = (1 + g) Z g z are unbiased estimators for X and Z. It is easy to see tat t x = N x k = t x and t z = N z k =. Correlation coefficient between variables y and x, Corr(y, x ) = Corr(y, x) = ρ yx and Corr(y, z ) = ρ yz. Te dual to ratio-cum-product estimator is ˆt RP = N ȳ x X Z z. Te approximate variance of ˆt RP is MSE(ˆt RP ) = N 2 1 f n Ȳ 2( C 2 y + gc 2 z (g + 2K yz) + gc 2 x(g 2gK zx 2K yx) ), It is sown in Sing et al (2005) tat for 1 g > 0 (N > 2n), ˆt RP Tompson estimator ˆt y and ratio-cum-product estimator ˆt RP wen is more efficient tan Horvitz- 1 2 g < Kyxc2 x K yzc 2 z < 1 (1 + g). c 2 x + c 2 z 2K zxc 2 x Stratified simple random sample case Consider stratified simple random sampling of size in stratum U, denote g = /( ) for = 1,..., H. Te dual transformation for stratified and arbitrary sampling design is defined in Plikusas (2008).

4 Here we use te direct generalization of dual transformation, and define transformation of te auxiliary variable x: x k = (1 + g ) X g x k, for k U, X = 1 k U x k. Te transformation for te variable z are defined analogously. Note tat N x k sample. = N x k = t x. Te relation Corr(y, x ) = ρ xy is not valid in te case of stratified Te dual to ratio-cum-product estimator is defined as ˆt yst = ˆt RP st = ˆt yst ˆt xst t x ˆt zst and ˆt xst =, Te approximate variance of dual to ratio-cum-product estimator is MSE(ˆt RP st) = x k. Ȳ 2( Cy 2 + g Cz(g 2 + 2K yz ) + g Cx(g 2 2g K zx 2K yx ) ), And te estimator of te approximate variance MSE(ˆt RP st) = 4 Simulation study Ȳ 2( Ĉy 2 + g Ĉz(g n ˆK yz ) + g Ĉx(g 2 2g ˆKzx 2 ˆK yx ) ). In tis section some empirical study is presented to observe te beavior of te estimators in te case of stratified simple random sample design. A real populations from some Lituanian Enterprise survey were used for te simulation. During te simulation study several populations were examined. It was observed tat for skewed populations, and big sampling fractions from strata, te product estimators are not efficient. Tey are beaten by simple ratio estimator, despite te ig correlation wit bot variables. It sould be noted tat bot auxiliary variables initially are positively correlated wit te study variable. So, first of all we transform te variable z to dual, and consider te transformed variable as given negatively correlated auxiliary variable. Below some results wen product estimators performs efficiently are presented. Population I y - An income of enterprise, x - Number of employees, z - Number of employees from anoter source (dual variable). N = 150, n = 50, t y = , t x = 11875, = 20433, C 2 y = , C 2 x = , C 2 z = , ρ yx = , ρ yz = , ρ zx = Population II y - An income of enterprise, x - Number of employees, z - Number of employees from anoter source

5 (dual variable). N = 150, n = 50, t y = , t x = 8659, = 13581, Cy 2 = , Cx 2 = , Cz 2 = , ρ yx = , ρ yz = , ρ zx = Tese populations are stratified into tree strata by te size of te variable x. Te sample size in eac strata satisfies inequality 2 < and 100 samples were drawn. Table 1. Simulation results for te Population I Esti- Average Esti- Average Approximator esti- mated estimate mate MSE mate bias of variance variance CV ˆt yst ˆt Rst ˆt RP st ˆt RP st Table 2. Simulation results for te Population II Esti- Average Esti- Average Approximator esti- mated estimate mate MSE mate bias of variance variance CV ˆt yst ˆt Rst ˆt RP st ˆt RP st Tables 1 and 2 sow tat for stratified simple random sampling design dual ratio-cum-product estimator can be more efficient tan oter estimators considered. For Population II (Table 2), dual ratio-cumproduct estimator as a little bit bigger coefficient of variation ten ratio-cum-product estimator. References Bandyopadyay, S. (1980) Improved ratio and product estimators. Sankyā Series C, 42(2), Plikusas, A. (2008) Some overview of te ratio type estimators In: Worksop on survey sampling teory and metodology, Statistics Estonia. Sing, M. P. (1969) Comparison of some ratio-cum-product estimators. S ankyā Series B, 31, Sing, H. P., Sing, R., Espejo, M. R., Pineda, M. D., Nadarajan, S. (2005) On te efficiency of te dual to ratio-cum-product estimator. Matematical Proceedings of te Royal Iris Academy, 105A(2),

6 Srivenkataramana, T. (1980) A dual to ratio estimator in sample surveys. Biometrica, 67, Upadyaya, L. N., Sing, H. P. (1999) Use of transformed auxiliary variable in estimating te finite population mean. Biometrical Journal, 41(5), Upadyaya, L. N., Sing, H. P., Vos, J. V. E. (1985) On te estimation of population means and ratios using supplementary information. Statistica Neerlandica, 39(3),