1. Introduction

In survey sampling, statisticians are often interested in dealing with variables that have highly skewed distributions such as consumptions, expenditure, income, etc. In such situations median is regarded as a more appropriate measure of location than mean. It has been well recognized that use of auxiliary variable 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), Mak and Kuk (1993), Garcia and Cebrian (2001), Singh et al. (2006), Al and Cingi (2010) used the auxiliary information in median estimation. Singh and Solanki (2013), proposed the four estimators and a generalized class of estimators of population median using known information of population median of the auxiliary variable. Empirically, they studied the 22 estimators.

In this paper, we first establish the new parametric relationship for population median and propose three estimators of under different situations. This technique of estimation of population median is different from the existing one given by Kuk and Mak (1989), which is very simple and efficient. Up to terms of order the optimum biases and minimum MSE’s of the proposed estimators are obtained and compared with each other. Empirically, we have done two studies, first to check the authenticity of the relationship for and to have the rough idea about the efficiencies of proposed estimators with each other. Secondly, to compare the efficiencies of the proposed estimators with existing ones as the techniques of estimation and parameters involved in their minimum MSE’s are different.

2. Notations and Results

Suppose a simple random sample of size is drawn from a finite population of size without replacement and observations on both study variables and auxiliary variable are taken. Let the values of variable and be denoted by and respectively on the unit of the population and the corresponding small letters and denote the sample values corresponding to the unit in the sample.

Taking,

Obviously

Defining,

For the sake of simplicity, assume that is large enough as compares to so that finite population correction (fpc) terms are ignored throughout.

For the given SRSWOR, we have the following expectations,

and up to terms of order

3. Proposed Estimators and Their Biases and MSE’s

Sharma et al. (2016) established the parametric relation for population mode as

where is unknown constant to be determined.

Sharma et al. (2016) also proposed three estimators of under the three different situations as

They determined the optimum value of and by minimizing the MSE’s of respective estimators as

where,

Note that are further functions of unknown population parameters. Srivastava and Jhajj (1983) have shown that if we replace the parameters involved in the optimum value of the unknown constant by their consistent estimators then up to terms of order the MSE remains the same. Therefore, the presence of unknown parameters in the optimum values of the constants will not create any problem for practical use of the proposed estimators.

Let be the values when parameters are replaced by their consistent estimators in So then the estimators reduces to

For moderately skewed distribution, Doodson (1917) suggested the empirical relationship as

which is also known as Karl Pearson empirical relation between mean, median and mode.

That implied

or

Using above, obviously, we have

(3.1)

where for univariate population, and for the bivariate population is highly positive and is highly negative, which is a new parametric relationship for population median.

Using above parametric relationship for population median, we here propose three different estimators of as

(3.2)

(3.3)

and

(3.4)

Up to terms of order the biases and MSE’s of and are given as

The biases of above estimators are of order so their contribution to the MSE’s will be of the order

Up to terms of order the MSE’s are

Among the three estimators, the most efficient is

4. Empirical and Simulation Studies

4.1. Empirical Studies

To illustrate the result numerically, we have made computations for 10 populations taken from literature, by using Microsoft Excel 2010.

Empirical study-I

The source of the populations, the nature of the variables, the values of and are listed in Table 1.

Table 1. Description of populations

In Table 2, we have given the true value of population Median and values of the population median obtained by using the new parametric functional relationship of and the efficiencies of proposed estimators are given in Table 3.

Table 2. Values of population median       and their values obtained from new parametric relationship

Table 3. Up to terms of       MSE of       and

From Table 2, we observe that the values obtained from the new parametric functional relationship for median i.e. are very close to the true value of the population median The small amount of errors is due to the fact that we are using the optimum value of (by minimizing MSE up to order not the exact one. So, empirically, this relation is verified and the percentage of relative margin of these errors presented in bar graph.

From Table 3, we observe that in the sampling theory, for the bivariate population, the known value of increases the accuracy of the estimators of population median We note that the MSE’s of all and are less than at the same time MSE’s of and are less than and respectively when conditions (i), (ii) and (iii) are satisfied and the relative efficiencies of and are also shown through bar graph.

Empirical Study-II

Theoretically, the proposed estimators of cannot be compared with the existing estimators because the techniques of estimators of and the parameters involved in MSE’s are different. So an empirical study has been done to check the performance of the proposed estimators over the existing ones, we consider the two populations, which are considered by Singh and Solanki (2013).

The minimum MSE’s of proposed estimators and estimators considered by Singh and Solanki (2013) are given in Table 4.

Table 4. MSE and Relative Efficiencies of Population Median

From Table 4, we observe that, the efficiency of the proposed estimator (appropriate estimator because is highly positive), which is very much high as compared to existing ones.

4.2. Simulation Study

A simulation study as been carried out using the software that verifies the theoretical results of our work. For this purpose, 100,000 samples have drawn from different gamma distributions, which are skewed. In this case the actual value of is used which is equal to 0.5. For these distributions, we check the behavior of the estimator for different sample sizes. The values of population mean, median and mode for different gamma distributions are shown in Table 5.

Table 5. Descriptive Statistics

Using simulation study we compared the existing estimator of sample median with the proposed estimator The expected values and MSE’s of these estimators are shown in Table 6.

Table 6. The Expected and MSE values of the Estimators

Some results based on simulation study

1. Biases and MSE’s for both the estimators decrease as the sample size in- creases.

2. The proposed estimator slightly more biased than the existing estimator.

3. Variance of proposed estimator always less than the variance of existing estimator, which shows that the proposed estimator is more stable than the existing estimator (sample median).

5. Conclusions

Theoretically and the with the support of empirical and simulation studies, we can say that the developed new parametric relationship for population median i.e. is true. Using this new parametric relationship for population median, we can construct the different estimators for population median. This different technique of estimation of population median is very simple and efficient as compared to the existing ones. Further, one more interesting conclusion from this study is that the proposed conventional consistent estimator of population median is more stable than the mean per unit estimator of population mean.