New Efficient Estimators of Population Mean Using Non-Traditional Measures of Dispersion

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1. Introduction

Sampling is done when the population is very large and we have to get the result very soon. The population parameters are estimated by the corresponding statistics in a natural sense. As it has been mentioned that the most suitable estimator for the estimation of population parameter is the corresponding statistics so to estimate population mean the most suitable estimator is the sample mean. Although he sample mean is an unbiased estimator of population mean and it has reasonably large variance and our aim is to search for the estimator with minimum variance or may be biased but with minimum mean squared error. This purpose is solved through the use of auxiliary information. Auxiliary information is obtained from auxiliary variable which is highly positively or negatively correlated with main variable under study. When the auxiliary variable is positively correlated with the main variable under study, ratio type estimators are used for improved estimation of population parameters. When it is negatively with the main variable under consideration, product type estimators are used for improved estimation of population parameters. In the present manuscript, we have confined our study to positively correlated populations only and proposed three ratio type estimators for improved estimation of population mean with higher efficiencies.

Let the population under consideration consists of N distinct and identifiable units and let $\left({x}_{i},{y}_{i}\right),i=1,2,\cdots ,n$ be a two variable sample of size n taken from bivariate variables (X, Y) through simple random sampling without sampling scheme. Let $\stackrel{\xaf}{X}$ and $\stackrel{\xaf}{Y}$ be the population means of the auxiliary and the study variables respectively, and let $\stackrel{\xaf}{x}$ and $\stackrel{\xaf}{y}$ be the respective sample means and both are unbiased estimators of $\stackrel{\xaf}{X}$ and $\stackrel{\xaf}{Y}$ respectively. Let the correlation coefficient between the variables X and Y be denoted by $\rho $ .

2. Existing Estimators under Review

As mentioned above most appropriate estimator of population mean is the sample mean $\stackrel{\xaf}{y}$ given by,

${t}_{o}=\stackrel{\xaf}{y}=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{\sum}}{y}_{i}}$

The above estimator is unbiased for population mean of the study variable and its variance up to the first order of approximation is given by,

$V\left({t}_{0}\right)=\frac{1-f}{n}{S}_{y}^{2}$ (1)

Cochran [2] proposed the following usual ratio estimator of population mean by using positively correlated auxiliary variable as,

${t}_{R}=\stackrel{\xaf}{y}\frac{\stackrel{\xaf}{X}}{\stackrel{\xaf}{x}}$

This estimator is biased and the bias and mean squared error of this estimator, up to the first order of approximation respectively are given by,

$B\left({t}_{R}\right)=\frac{1-f}{n}\frac{1}{\stackrel{\xaf}{X}}\left[{R}_{1}{S}_{x}^{2}-\rho {S}_{y}{S}_{x}\right]$

$MSE\left({t}_{R}\right)=\frac{1-f}{n}\left[{S}_{y}^{2}+{R}_{1}^{2}{S}_{x}^{2}-2{R}_{1}\rho {S}_{y}{S}_{x}\right]$ , (2)

where ${R}_{1}=\frac{\stackrel{\xaf}{Y}}{\stackrel{\xaf}{X}}$

Many estimators of population mean have been given by various authors in the literature for improved estimation. The latest references can be made of Yadav [3] , Yadav and Kadilar [4] [5] , Yadav et al. [6] [7] [8] [9] , Yadav and Mishra [10] , Misra and Gupta [11] [12] and Misra et al [13] . The Table 1 below represents different estimators of population mean using auxiliary variable along with their constants, biases and their mean squared errors.

Table 1. Various estimators of population mean, bias, mean squared error and constant.

3. Proposed Estimators

Motivated by Abid et al. [1] and Subramani [19] and searching for the improved estimators, we have used a specific parameter as the ratio of correlation coefficient and coefficient of skewness of auxiliary variable along with some non-tra- ditional parameters of auxiliary variable given by Abid et al. [1] as,

${t}_{{p}_{1}}=\frac{\stackrel{\xaf}{y}+b\left(\stackrel{\xaf}{X}-\stackrel{\xaf}{x}\right)}{\left(\tau \stackrel{\xaf}{x}+G\right)}\left(\tau \stackrel{\xaf}{X}+G\right),$

${t}_{{p}_{2}}=\frac{\stackrel{\xaf}{y}+b\left(\stackrel{\xaf}{X}-\stackrel{\xaf}{x}\right)}{\left(\tau \stackrel{\xaf}{x}+D\right)}\left(\tau \text{}\stackrel{\xaf}{X}+D\right),$

${t}_{{p}_{3}}=\frac{\stackrel{\xaf}{y}+b\left(\stackrel{\xaf}{X}-\stackrel{\xaf}{x}\right)}{\left(\tau \stackrel{\xaf}{x}+{S}_{pw}\right)}\left(\tau \stackrel{\xaf}{X}+{S}_{pw}\right),$

where, $\tau =\rho /{\beta}_{1}$

To study the large sample approximations, we have used the following approximations as,

$\stackrel{\xaf}{y}=\stackrel{\xaf}{Y}\left(1+{e}_{0}\right)$ and $\stackrel{\xaf}{x}=\stackrel{\xaf}{X}\left(1+{e}_{1}\right)$

such that

$E\left({e}_{i}\right)=0,\text{\hspace{0.17em}}i=0,\text{\hspace{0.17em}}1$

and

$E\left({e}_{0}^{2}\right)=\frac{1-f}{n}{C}_{y}^{2}$ , $E\left({e}_{1}^{2}\right)=\frac{1-f}{n}{C}_{x}^{2}$ ,

and

$E\left({e}_{0}{e}_{1}\right)=\frac{1-f}{n}{C}_{yx}=\frac{1-f}{n}\rho {C}_{y}{C}_{x}$ ,

where $f=\frac{n}{N}$ , ${C}_{y}^{2}=\frac{{S}_{y}^{2}}{{\stackrel{\xaf}{Y}}^{2}}$ , and ${C}_{x}^{2}=\frac{{S}_{x}^{2}}{{\stackrel{\xaf}{X}}^{2}}$ .

Using above approximation and up to the first order of approximations, the biases and the mean squared errors of proposed estimators are given by,

$B\left({t}_{{p}_{j}}\right)=\frac{1-f}{n}\frac{{S}_{x}^{2}}{\stackrel{\xaf}{Y}}{R}_{{p}_{j}}^{2},\text{\hspace{0.17em}}\left(j=1,2,3\right)$

$MSE\left({t}_{{p}_{j}}\right)=\frac{1-f}{n}\left[{R}_{{p}_{j}}^{2}{S}_{x}^{2}+{S}_{y}^{2}\left(1-{\rho}^{2}\right)\right],\text{\hspace{0.17em}}\left(j=1,2,3\right)$ (3)

where,

${R}_{{p}_{1}}=\frac{\stackrel{\xaf}{Y}\tau}{\stackrel{\xaf}{X}\tau +G}$ , ${R}_{{p}_{2}}=\frac{\stackrel{\xaf}{Y}\tau}{\stackrel{\xaf}{X}\tau +D}$ , ${R}_{{p}_{3}}=\frac{\stackrel{\xaf}{Y}\tau}{\stackrel{\xaf}{X}\tau +{S}_{pw}}$

4. Efficiency Comparison

In this section, the proposed estimators have been compared theoretically with the other existing estimators of population mean in terms of theirs variances and mean squared errors under simple random sampling without replacement scheme.

From Equation (3) and the from the Equation (1), the proposed estimators performs better than the mean per unit estimator if,

$MSE\left({t}_{{p}_{j}}\right)-V\left(\stackrel{\xaf}{y}\right)\le 0$

or,

$\left[{R}_{{p}_{j}}^{2}{S}_{x}^{2}-{\rho}^{2}{S}_{y}^{2}\right]\le 0$

or,

${R}_{{p}_{j}}^{2}\le \frac{{\rho}^{2}{S}_{y}^{2}}{{S}_{x}^{2}}$

or,

${R}_{{p}_{i}}\le \pm \frac{\rho {S}_{y}}{{S}_{x}},\text{\hspace{0.17em}}\left(j=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3\right)$ (4)

The proposed estimators ${t}_{{p}_{j}}\left(j=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3\right)$ in Equation (3) are better than the ratio estimator by Cochran [2] ${t}_{r}$ in Equation (2) under the condition if,

$MSE\left({t}_{{p}_{j}}\right)-MSE\left({t}_{r}\right)\le 0$

or,

$\left[\left({R}_{{p}_{j}}^{2}-{R}_{1}^{2}\right){S}_{x}^{2}-{\rho}^{2}{S}_{y}^{2}+2{R}_{1}\rho {S}_{y}{S}_{x}\right]\le 0$

or,

$\left({R}_{{p}_{j}}^{2}-{R}_{1}^{2}\right){S}_{x}^{2}\le {\rho}^{2}{S}_{y}^{2}-2{R}_{1}\rho {S}_{y}{S}_{x},\text{\hspace{0.17em}}\left(j=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3\right)$ (5)

From Equation (3) and the mean squared error of the estimators given by Kadilar and Cingi [14] in Table 1, the proposed estimators perform better than the Kadilar and Cingi [14] estimators under the condition if,

$MSE\left({t}_{{p}_{j}}\right)-MSE\left({t}_{i}\right)\le 0$

or,

$\left[{R}_{{p}_{j}}^{2}{S}_{x}^{2}-{R}_{i}^{2}{S}_{x}^{2}\right]\le 0$

or,

${R}_{{p}_{j}}\le \pm {R}_{i},\text{\hspace{0.17em}}\left(j=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3\right),\text{\hspace{0.17em}}\left(i=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,4,\text{\hspace{0.17em}}5\right)$ (6)

From the mean squared errors of proposed estimators and Kadilar and Cingi [15] estimators respectively in Equation (3) and in Table 1, the proposed estimators are better than the Kadilar and Cingi [15] estimators if,

$MSE\left({t}_{{p}_{j}}\right)-MSE\left({t}_{i}\right)\le 0$

or,

$\left[{R}_{{p}_{j}}^{2}{S}_{x}^{2}-{R}_{i}^{2}{S}_{x}^{2}\right]\le 0$

or,

${R}_{{p}_{j}}\le \pm {R}_{i},\text{\hspace{0.17em}}\left(j=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3\right),\left(i=6,\text{\hspace{0.17em}}7,\text{\hspace{0.17em}}8,9,\text{\hspace{0.17em}}10\right)$ (7)

From Equation (3) and the mean squared error of the estimators given by Yan and Tian [16] in Table 1, the proposed estimators are better than Yan and Tian [16] estimators if,

$MSE\left({t}_{{p}_{j}}\right)-MSE\left({t}_{i}\right)\le 0$

or,

$\left[{R}_{{p}_{j}}^{2}{S}_{x}^{2}-{R}_{i}^{2}{S}_{x}^{2}\right]\le 0$

or,

${R}_{{p}_{j}}\le \pm {R}_{i},\text{\hspace{0.17em}}\left(j=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3\right),\text{\hspace{0.17em}}\left(i=11,\text{\hspace{0.17em}}12\right)$ (8)

From Equation (3) and the mean squared errors of the estimators given by Subramani and Kumarpandiyan [17] in Table 1, the proposed estimators perform better than Subramani and Kumarpandiyan [17] estimators if,

$MSE\left({t}_{{p}_{j}}\right)-MSE\left({t}_{i}\right)\le 0$

or,

$\left[{R}_{{p}_{j}}^{2}{S}_{x}^{2}-{R}_{i}^{2}{S}_{x}^{2}\right]\le 0$

or,

${R}_{{p}_{j}}\le \pm {R}_{i},\text{\hspace{0.17em}}\left(j=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3\right),\text{\hspace{0.17em}}\left(i=13,\text{\hspace{0.17em}}14,\text{\hspace{0.17em}}15,\text{\hspace{0.17em}}16\right)$ (9)

The proposed estimators are better than the estimators by Jeelani et al. [18] in Table 1 under the condition if,

$MSE\left({t}_{{p}_{j}}\right)-MSE\left({t}_{17}\right)\le 0$

or,

$\left[{R}_{{p}_{j}}^{2}{S}_{x}^{2}-{R}_{17}^{2}{S}_{x}^{2}\right]\le 0$

or,

${R}_{{p}_{j}}\le \pm {R}_{17},\text{\hspace{0.17em}}\left(j=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3\right)$ (10)

From MSE of the proposed estimators in Equation (3) and the estimators given by Abid et al. [1] , it is found the proposed estimators are better than Abid et al. [1] estimators if,

$MSE\left({t}_{{p}_{j}}\right)-MSE\left({t}_{i}\right)\le 0$

or,

$\left[{R}_{{p}_{j}}^{2}{S}_{x}^{2}-{R}_{i}^{2}{S}_{x}^{2}\right]\le 0$

or,

${R}_{{p}_{j}}\le \pm {R}_{i},\text{\hspace{0.17em}}\left(j=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3\right),\text{\hspace{0.17em}}\left(i=18,\text{\hspace{0.17em}}19,\text{\hspace{0.17em}}\cdots ,\text{\hspace{0.17em}}26\right)$ (11)

5. Empirical Example

To judge the performances of the proposed and the existing estimators of population mean and to verify the conditions under which proposed estimators performs better than the existing estimators, we have considered the population given by Kadilar and Cingi [14] . The numerical values of the constants, biases and the mean squared error of the proposed and the existing estimators have been calculated for this data. The population parameters for the above population are as follows:

$N=106$ , $n=40$ , $\stackrel{\xaf}{Y}=2212.59$ , $\stackrel{\xaf}{X}=27421.70$

$\rho =0.860$ , $\rho =0.860$ , ${C}_{y}=5.22$ , ${S}_{x}=57460.61$

${C}_{x}=2.10$ , ${\beta}_{1}=2.122$ , ${\beta}_{2}=34.572$ , ${M}_{d}=7297.50$

$QD=12156.25$ , $G=40201.69$ , $D=35634.99$ , ${S}_{pw}=35298.81$

Table 2 represents the numerical values of constants, biases and the mean squared errors of proposed and other existing estimators of population mean using auxiliary variable for the above data.

6. Results

Form Table 2, we see that the proposed estimators are having lesser biases and mean squared errors as compared to all existing estimators. So the proposed estimators are more efficient than the other estimators for estimating population mean. Our purpose to search for the estimator with higher efficiency is achieved.

Table 2. Constants, Biases and MSE of Proposed and other estimators.

Further it is to be mentioned that among the proposed estimators, ${t}_{{p}_{1}}$ is the best as it has smallest bias and the mean squared error.

7. Conclusion

This paper deals with the estimation of population mean of the study variable using auxiliary variable in the form of a special parameter along with some non-traditional measures of dispersion of auxiliary variable used by Abid et al. [1] . The expressions for the biases and mean squared errors of these proposed estimators have been derived up to the first order of approximation. A theoretical comparison of the proposed estimators has been made with the existing estimators of population mean under simple random sampling scheme. An empirical study is also carried out to judge the performances of the proposed and existing estimators of population mean. Through this numerical study, it has been found that the proposed estimators are more efficient than the other existing estimators. As proposed estimators are more efficient estimators for population mean, so they should be used for the improved estimation of population mean of study variable using auxiliary variable under simple random sampling scheme.

Acknowledgements

The authors are thankful to editor of Open Journal of Statistics and anonymous referees for critically examining the manuscript which helped in improving the earlier draft.

Notations

The following given by Abid [1] have been used in this manuscript and are as,

$N$ - Size of the population,

$n$ - Size of the sample,

$Y$ - Study variable,

$X$ - Auxiliary variable,

$\stackrel{\xaf}{Y},\text{\hspace{0.17em}}\stackrel{\xaf}{X}$ - Population means,

$\stackrel{\xaf}{y},\stackrel{\xaf}{x}$ - Sample means,

${S}_{y},{S}_{x}$ - Population Standard Deviations,

${S}_{yx}$ - Population Covariance between Y and X,

${C}_{y},{C}_{x}$ - Coefficients of Variation,

${M}_{d}$ - Median of the auxiliary variable,

$\rho $ - Correlation coefficient between X and Y,

$b=\frac{{s}_{yx}}{{s}_{x}^{2}}$ - Regression coefficient of y on x,

${\beta}_{1}=\frac{N{\displaystyle \underset{i=1}{\overset{N}{\sum}}{\left({X}_{i}-\stackrel{\xaf}{X}\right)}^{3}}}{\left(N-1\right)\left(N-2\right){S}_{x}^{3}}$ - Coefficient of Skewness of auxiliary variable,

${\beta}_{2}=\frac{N\left(N+1\right){\displaystyle \underset{i=1}{\overset{N}{\sum}}{\left({X}_{i}-\stackrel{\xaf}{X}\right)}^{4}}}{\left(N-1\right)\left(N-2\right)\left(N-3\right){S}_{x}^{4}}-\frac{3{\left(N-1\right)}^{2}}{\left(N-2\right)\left(N-3\right)}$ - Coefficient of Kurtosis of auxiliary variable,

$QD=\frac{{Q}_{3}-{Q}_{1}}{2}$ - Quartile Deviation,

$G=\frac{4}{N-1}{\displaystyle \underset{i=1}{\overset{N}{\sum}}\left(\frac{2i-N-1}{2N}\right){X}_{i}}$ - Gini’sMean Difference,

$D=\frac{2\sqrt{\text{\pi}}}{N\left(N-1\right)}{\displaystyle \underset{i=1}{\overset{N}{\sum}}\left(i-\frac{N+1}{2}\right){X}_{i}}$ - Downton’s Parameters,

${S}_{pw}=\frac{\sqrt{\text{\pi}}}{{N}^{2}}{\displaystyle \underset{i=1}{\overset{N}{\sum}}\left(2i-N-1\right){X}_{i}}$ - Probability Weighted Moments,

$B(.)$ - Bias of the estimator,

$V(.)$ - Variance of the estimator,

$MSE(.)$ - Mean squared error of the estimator,

$PRE\left({t}_{e},{t}_{p}\right)=\frac{MSE\left({t}_{e}\right)}{MSE\left({t}_{p}\right)}\ast 100$ - Percentage relative efficiency of the estimator ${t}_{p}$ over ${t}_{e}$ .

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