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Efficient Estimation of Population Median Using Supplementary Variable

In this paper, we propose an exponential cum ratio-product type class of estimators for population median under simple random sampling scheme using the supplementary variable. Expressions for bias and mean square error (MS E) are obtained up to first order of approximation. The proposed class of estimators is more efficient as compared to all considered estimators under certain conditions. Four real data sets and simulation studies are carried out to observe the performances of the estimators. Both numerical and simulation studies show that the proposed class of estimators performs better as compared to all considered estimators.


Introduction
When the supplementary variable is correlated with the study variable, then in such situation, the use of the appropriate supplementary variable may results in considerable reduction in the mean squared error (MSE) of the regression, ratio and product estimators. A primary objective in many statistical analysis is to estimate the location parameter for the desired distribution, which describes the data in good manners. When the entire data is homogeneous, the sample mean performs better. In survey sampling, there are situations when the study variable follows extremely skewed distribution, such as expenditure, taxes, income, production and consumption or in data, if some of the observations lie very far from 1 rest of the observations, then in such situation, the sample mean performs very poorly and the results are either under or over estimated. To handle this kind of situation, median is used as an alternative measure of location, which performs better in case of skewed variable and gives efficient estimate for population parameter. A large amount of literature is available to estimate the mean and total, but comparatively less efforts have been devoted to propose efficient estimators for population median (M y ). Gross [1] was the first who proposed a weighted estimator without the supplementary variable for population median (M y ) and derived its asymptotic distribution. Kuk and Mak [2] suggested the classical ratio method for population median using the supplementary information. Singh et al. [3] suggested a class of generalized ratio type estimators using the supplementary variable for population median. Singh et al. [4,5] suggested certain estimators for population median using known parameters of the supplementary variables. Singh et al. [6] suggested chain ratio and regression estimators for population median. Gupta et al. [7] proposed number of difference types estimators for population median using the supplementary variables. Other important contributions in this area include, Aladag and Cingi [8], Singh and Solanki [9], Jhajj et al. [10], Sharma and Singh [11], Aladag and Cingi [12], Shabbir and Gupta [13,14], Solanki and Singh [15], Enang et al. [16], Jhajj et al. [17], Koyuncu [18], Irfan et al. [19], Yadav et al. [20], Bandyopadhyay et el. [21], Javed et al. [22] and Baig et al. [23]. These researchers suggested large number of estimators for estimation of population parameters under different situations using the supplementary variables.
In this article, we propose a new class of exponential cum ratio-product type estimators for population median (M y ) under simple random sampling (SRS) scheme using the supplementary variable. The main aim of this research is to propose an efficient class of estimators to increase precision of the median estimators.
Consider a finite population U = {U 1 , U 2 , . . . U N } consisting of N units. Let (y i , x i ) be the ith values of the survey variable y and the supplementary variable x respectively. From this population, a sample of size n is selected using simple random sample without replacement (SRSWOR) scheme. Let M y and M x be the sample medians corresponding to population medians M y and M x respectively. Let f y (M y ) and f x (M x ) be the probability density functions of M y and M x respectively. Let ρ ( M y , M x ) = ρ yx = 4P 11 (y, x) − 1 be the coefficient of correlation between M y and M x , where P 11 indicates the fraction of values in the population 2 with Y ≤ M Y and X ≤ M X . It is assumed that (y, x) follow a continuous distribution with f y (M y ) and f x (M x ) being their marginal densities.
To get the bias and MSEs of the estimators, we use the following relative error terms.

Estimators in Literature
We discuss the following usual median per unit estimator and some existing estimators of population median M y .
(i). The most common estimator of median M y is sample median estimator M y whose variance, is given by (ii). The usual ratio estimator of median suggested by Kuk and Mak [2] as: (iv). Singh and Solanki [9] proposed the following estimators for M y as: where k 1 , k 2 and k 3 are constants. The biases and minimum MSEs of M S S i (i = 1, 2), to first degree of approximation, are given by .
The optimal values of k j ( j = 1, 2, 3) are: The difference estimator of median as: where k 4 is unknown constant, whose value is to be find. The minimum MSE of M D , is given by The optimum value of . Solanki and Singh [15] proposed a generalized class of estimators for median as: where k 5 and k 6 are unknown constants and c, d are known population parameters of the supplementary variable. The bias and minimum MSE of M S S 3 , at (c, d) = (1, 0), are given by and The optimum values of k 5 and k 6 are (vii). Shabbir and Gupta [14] suggested the following difference type estimators of median as: where k j ( j = 7, 8, . . . , 10) are constants. The biases and minimum MSEs of M Di (i = 1, 2), are given by The optimal values of k j ( j = 7, 8, 9, 10) are:

The Proposed Estimator
Motivated by Solanki and Singh [15] and Muneer et al. [24], we propose the following class of exponential cum ratio-product type estimators of population median M y as where (k j , j = 11, 12) are unknown constants whose values are to be determined and α is known constant which takes values (0, 1). Putting the values of α in Equation (25), we get the following two estimators.
(i) Put α = 0, the proposed estimator is reduced to: (ii) Put α = 1, the proposed estimator is reduced to: Expressing M Prop in Equation (25) in terms of (ζ j , j = 0, 1) upto first order of approximation, we get 6 Keeping ζ s upto power two and expanding Equation (28), we have Solving Equation (29), the bias and MSE of M Prop are: or MS E( M Prop ) M 2 y 1 + k 2 11 A 1 + k 2 12 A 2 + 2k 11 k 12 A 3 − 2k 11 A 4 − 2k 12 A 5 , (31) where Differentiate Equation (31) with respect to k 11 and k 12 , we obtain the optimal values of k 11 and k 12 as: Substitute the optimal values of k 11 and k 12 in Equation (31), we get the minimum MSE of M Prop , is given by: (32)

Theoretical Comparison
Now we compare the MSE of the proposed estimatorM Prop with the MSEs of considered estimators and derive the optimal conditions. Condition (i): By (1) and (32), (4) and (32), (7) and (32),

Numerical Comparisons
To examine performances of the estimators, we use the following data sets for comparisons. The descriptive statistics of all populations are given in Table 1.

MS Es and
PREs values based on four populations are given in Table 2.
From Table 2, it is clear that the MS E of the proposed class of estimators in all four populations have minimum as compared to considered estimators. It is also observed that using different correlations, the proposed estimators are more efficient and stable. 10

Simulation Study
The simulation study is carried out to observe the performances of the estimators both empirically and theoretically. We consider three finite populations of size 1500 generated from a skewed multivariate normal distribution with same theoretical means of [Y, X] as µ = [5,5] and using different variance covariance matrices as given below. For each population, we consider sample sizes n = 50, 100, 150, 300 and probability density functions f y (M y ) and f x (M x ) are calculated using normal distribution. In Table 3, the empirical and theoretical mean squared errors and percent relative efficiencies are computed. We estimate the empirical MSE using 5000 samples of various sizes selected from each population.
Analyzing the simulation results of all three artificial populations in Table  3, it is clear that the proposed class of estimators performs more efficiently and have less MS E as compared to all considered estimators in all populations. It is noted that the empirical MS E ofM S S 3 is very high in all populations and performs very poorly.

Conclusion
In this article, we have proposed a new class of exponential cum ratio-product type estimators for the population median (M y ) under simple random sampling using the supplementary variable. The bias and MSE expressions are obtained upto first order of approximation. Theoretical conditions are derived under which the proposed class of estimators performs better. In Tables 2 and 3     Ghulam Narjis as completed her Ph.D degree in Statistics from Quaid-i-Azam University Islamabad, Pakistan. She has published many articles in internationally reputed journals in the field of survey methodologies, randomized response and data analysis. Her area of interest include Sampling Theory and Bayesian Analysis.