1. Introduction

Use of auxiliary information has been in practice to increase the efficiency of the estimators. When population parameters of the auxiliary variable are known, several estimators for population mean of study variable have been discussed in the literature. When the variable under study and the auxiliary variables are highly and positively correlated and the line of regression passes through origin, ratio method of estimation is preferred to use. On the other hand, if they are highly and negatively correlated, product method of estimation is suggested to use.

Let the finite population consists of N units. Suppose two auxiliary variables and are observed on where is positively and is negatively correlated with the study variable A simple random sample of size n with n < N, is drawn using simple random sampling without replacement (SRSWOR) from the population U to estimate the population mean of study character when the population means and of and respectively are known.

Usual ratio and product estimators given by Cochran [1] and Robson [12] respectively for estimating the population mean respectively are defined as,

(1.1)

(1.2)

Upadhyaya and Singh [18] suggested ratio and product estimators utilizing coefficient of variation and coefficient of kurtosis of auxiliary variables as

(1.3)

(1.4)

(1.5)

(1.6)

To estimate , Singh [16] suggested a ratio-cum-product estimator as

(1.7)

Singh and Tailor [15] suggested a ratio-cum-product estimator of utilizing the correlation coefficient between auxiliary variables as

(1.8)

Tailor et.al [17] suggested two estimators of population mean using coefficients of variation and coefficients of kurtosis of auxiliary variables as,

(1.9)

(1.10)

To the first degree of approximation the mean squared error (MSE) of the estimators and respectively are

(1.11)

(1.12)

(1.13)

(1.14)

(1.15)

(1.16)

(1.17)

(1.18)

(1.19)

(1.20)

where

Many authors, such as Kadilar and Cingi ([2], [3]), Shabbir and Gupta [13], Singh and Vishwakarma [14], Koyuncu and Kadilar ([4], [5], [6], [7]), Sanaullah et al. [8], Mouhamed et al. [9], Parmar et al. [11], Tailor et al [17], Yadav et al. ([19], [20]), Onyeka et al. [10] have improved the ratio and product estimators as given in (2.1) and (2.3 for the population mean of the study variable in the stratified random sampling.

2. Proposed Estimator

Making the use of auxiliary information more appropriately, assuming that the information on co-efficient of kurtosis and median of auxiliary variables and are known, we propose the estimator as,

(2.1)

To obtain the bias and mean square error of the proposed estimator, we assume that

such that and

Expressing the proposed estimator in terms of we get

(2.2)

where

Taking expectation on both sides of (2.2), we get

Substituting the values of and we get the bias of as

(2.3)

To find the mean squared error of the suggested estimator up to first degree of approximation square and take expectation on both sides of (2.2). That is,

After substituting the values of and we get the mean squared error of as

(2.4)

3. Efficiency Comparison

We know that the variance of sample mean in simple random sampling without replacement (SRSWOR) is,

(3.1)

From (1.11) – (1.20), (2.4) and (3.1) we have

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

4. Empirical Study

The performance of the proposed estimator is assessed with that of SRSWOR sample mean, traditional Ratio-cum-Product estimators, and existing modified estimators for a certain hypothetical population of size 30, that is given as below:

The population parameters of the auxiliary variables and the constants computed from the above populations are given below:

Table 4.1. The variance of SRSWOR sample mean, mean squared errors of the existing and proposed modified estimators

To see the performance of the proposed estimator in comparison to and , we calculated the percent relative efficiency of proposed estimator with respect to and The Percent relative efficiency (%) of the proposed estimator with respect to the existing estimators (e) has been computed as and is presented in Table 4.2.

Table 4.2. Percent relative efficiencies of proposed estimator over other estimators

From the values of Table 4.2, it is observed that the proposed estimator is more efficient than the usual unbiased estimator ratio estimator product estimator and other existing estimators with considerable gain in efficiency.

5. Conclusions

In this paper, a ratio-cum-product estimator for the estimation of finite population mean with known coefficient of kurtosis and median of auxiliary characters has been proposed. The bias and mean squared error of the proposed estimator are obtained and compared with that of the SRSWOR sample mean, ratio estimator, product estimator, Singh and Tailor [15] and Singh [16], Upadhyaya and Singh [18] and Parmar et.al [11] estimators. Further, we have derived the conditions for which the proposed estimator is more efficient than the existing estimators. We have also assessed the performance of the proposed estimator with that of the existing estimators for a hypothetical population. It is observed that the mean squared error of the proposed estimator is less than the mean squared errors of the existing estimators. Hence, we strongly recommend that the proposed estimator may be preferred over the existing estimators for the use of practical application.

ACKNOWLEDGMENTS

The authors are very much thankful to the Editor in-Chief of AJOR, and to the anonymous learned referees for their valuable suggestions regarding improvement of the paper.