Efficient Class of Estimators for Population Median Using Auxiliary Information Prayas Sharma and Rajesh Singh Department of Statistics, Banaras Hindu University, Varanasi, India. [email protected], [email protected] Abstract This article suggests an efficient class of estimators of population median of the study variable using an auxiliary variable. Asymptotic expressions of bias and mean square error of the proposed class of estimators have been obtained. Asymptotic optimum estimator has been investigated along with its approximate mean square error. We have shown that proposed class of estimator is more efficient than estimator considered by Srivastava (1967), Gross (1980), Kuk and Mak (1989) Singh et al. (2003b), Al and Chingi (2009) and Singh and Solanki (2013). In addition theoretical findings are supported by an empirical study based on two populations to show the superiority of the constructed estimators over others. Key words: Auxiliary Variable , Simple random sampling, Bias, Mean Square Error. 1. Introduction In the sampling literature, statisticians are often interested in dealing with variables that have highly skewed distributions such as consumptions and incomes. In such situations median is considered the more appropriate measure of location than mean. It has been well recognised that use of auxiliary information results in efficient estimators of population parameters. Initially, estimation of median without auxiliary variable was analyzed, after that some authors including Kuk and Mak (1989), Meeden (1995) and Singh et al. (2001) used the auxiliary information in median estimation. Kuk and Mak (1989), proposed the problem of estimating the population median M y of study variable Y using the auxiliary variable X for the units in the sample and its median M x for the whole population. Some other important references in this context are Chambers and Dunstan (1986), Rao et al. (1990), Mak and Kuk (1993), Rueda et al. (1998), Arcos et al. (2005), Garcia and Cebrian (2001), Singh et al. (2003a, 2006), Singh et al. (2007) and Singh and Solanki (2013). Let Yi and Xi (i =1,2,....N) be the values of the population units for the study variable Y and auxiliary variable X, respectively. Further suppose that yi and xi (i=1,2.....n) be the values of the units including in the sample say, sn of size n drawn by simple random sampling without replacement (SRSWOR) scheme. Kuk and Mak (1989) suggested a ratio estimator for estimating population median My of the study variable Y, assuming population median of auxiliary variable X, M x is known, given as ˆ ˆ ˆ Mr My Mx / Mx (1.1) ˆ ˆ where My (due to Gross 1980) and M x are the sample estimators of My and M x respectively. Suppose that y(1) , y(2) ,..... y (n ) are the y values of sample unites in ascending order. Further, suppose t be an integer satisfying Y(t) M y M (t1) and p=t/n be the proportion of y values in the sample that are less than or equal to the median value My, an unknown population ˆ parameter. If Q y t denote the t-quantile of Y then My Qy 0.5. Kuk and Mak (1989) defined a matrix of proportion (pij) is Y M y Y M y Total X M x p11 p 21 p .1 X M x p12 p 21 p.2 Total p1 p 2 1 Following Robson (1957) and Murthy (1964), the product estimator for population median M y is defined as ˆ ˆ ˆ Mp My Mx / Mx (1.2) The usual difference estimator for population median M y is given by ˆ ˆ ˆ Md My dMx Mx (1.3) ˆ where d is a constant to be determined such that the mean square error of Md is minimum. Singh et al. (2003) proposed the following modified product and ratio estimators for population median M y , respectively, as a Mˆ Mˆ Mˆ x (1.4) 1 y a M x and a M Mˆ Mˆ x (1.5) 2 y ˆ a M x where a is suitably chosen scalar. Srivastava (1967) type estimator for median estimation is given by M Mˆ Mˆ x (1.6) 3 y ˆ M x Reddy (1973,1974) and Walsh (1970)-type estimator is given by M Mˆ Mˆ x (1.7) 4 y ˆ M x M x M x Sahai and Ray (1980)-type estimator is given by M ˆ ˆ x M5 M y 2 (1.8) Mˆ x Vos (1980)- type estimator is given by Mˆ Mˆ wMˆ (1 w)Mˆ x 6 y y M x (1.9) M Mˆ wMˆ (1 w)Mˆ x 7 y y ˆ M x where w is suitably chosen scalar . ˆ All the estimators considered from (1.1) to (1.9) and conventional estimator My are members of the Srivatava (1971) and Srivastava and Jhajj (1981)-type class of estimators Mˆ ˆ (G) ˆ (G) ˆ x G M y : M y G M y , (1.10) M X where the function G assumes a value in a bounded closed convex subset Q R 2 , which contains the point (My ,1) and is such that GM y ,1 1 Using first order –order Taylor’s series expansion about the point M y ,1, we have ˆ (G) ˆ 1 My GMy ,1 My My G10 My ,1 G01 My ,1 O(n ) (1.11) Mˆ where U x . M x G And G M ,1 01 y U (My ,1) Using conditions, we have ˆ (G) ˆ 1 My My My My U 1G01 My ,1 O(n ) Or ˆ (G) ˆ My My My My U 1G01 My ,1 (1.12) ˆ (G) Squaring and taking expectations of both sides of (1.12),we get the MSE of My to the first order of approximation as VMˆ CovMˆ ,Mˆ MSEMˆ (G) VMˆ y G 2 Mˆ ,1 2 y x G 2 Mˆ ,1 y y M 2 01 y M 01 y x x Here as N , n then n / N f and we assumed that as N the distribution of (X, Y) approaches a continues distribution with marginal densities f x x and f y y of X and Y respectively. Super population model framework is necessary for treating the values of X and Y in a realization of N independent observation from a continuous distribution. It is also ˆ assumed that f x M x and f y M y are positive. Under these conditions, sample median M y is consistent and asymptotically normal (due to Gross, 1980) with mean M y and variance ˆ 2 2 VMy MyCy and ˆ 2 2 VM x M x C x ˆ ˆ CovMy ,Mx cMyMxCyCx 1 where 1 f / 4n , f n / N Cy Myf y My Cx M x f x M x and C 4p11 1 with p11 PM x ,M y goes from -1 to +1 as p11 increase from 0 to 0.5 ˆ (G) Substituting these values we get the MSE of My to the first degree of approximation as ˆ (G) 2 2 2 2 MSEMy MyCy Cx G01 My ,1) 2cCxCyMyG01 My ,1 The MSE is minimum when G 01 M y ,1 k c M y (1.18) C y where k c c . Cx ˆ (G) Thus, the minimum MSE of My is given by ˆ (G) 2 2 2 ˆ MSEmin My CyMy 1 c MSEmin Md (1.19) ˆ Which is equal to the minimum MSE of the estimator Md defined at (1.3). ˆ ˆ ˆ It is to be mentioned that minimum MSE’s of the estimators M r , Mp and M i i 1,2...7 are equal to MSE expression given in equation (1.19). It is obvious from (1.19) that the ˆ (G) estimators of the form My are asymptotically no more efficient than the difference estimator at its optimum value or the regression type estimator given as ˆ ˆ ˆ ˆ Mlr My dMx Mx (1.20) fˆ Mˆ where dˆ x x 4pˆ 1 ˆ ˆ 11 f y M y Singh and Solanki (2013) suggested following classes of estimators ˆ 1 ˆ ˆ Md d1My 1 d1 Mx Mx (1.21) ˆ 2 ˆ ˆ Md d1My d2 Mx Mx (1.22) ˆ 3 ˆ ˆ Md d1My d2Mx 1 d1 d2 Mx (1.23) M Mˆ 4 d Mˆ d M Mˆ x (1.24) d 1 y 2 x x ˆ M x where d 1 and d 2 are suitable constants to be determined such that MSE’s of the estimators considered in (1.21) to (1.24) are minimum, and are either real numbers or the functions of the known parameters of auxiliary variable X. Biases and minimum MSEs of the estimators considered in (1.21) to (1.24) are given as ˆ 1 BMd (d1 1)My (1.25) ˆ 2 BMd (d1 1)My (1.26) ˆ 3 BMd (d1 1)1 R)My (1.27) ˆ 4 2 2 2 BMd My d1 1 Cx kc d2RCx 1 (1.28) 2 1 RC2 R k MSE Mˆ 1 M 2 1 R 2 C2 x c (1.29) min d y x 1 C2 RC 2 R 2k y x c M 2 C2 1 2 ˆ 2 y y c MSEmin M d 2 2 (1.30) 1 C y 1 c 2 2 2 2 3 M y C y 1 c 1 R MSE Mˆ (1.31) min d 2 2 2 1 R C y 1 c 1 2 C2 M 2 C2 1 2 ˆ 4 x y y c MSEmin M d 2 2 2 2 (1.32) 1 C x C y 1 c 2.