An Estimator of Population Variance Using Auxiliary Information under General Sampling Design

Vyas Dubey* and Minal Uprety

School of Studies in Statistics, Pt. Ravishankar Shukla University, Raipur, (C.G.)

ABSTRACT:

The paper deal with a difference type estimator of population variance using auxiliary information is more efficient than various estimators under stringent conditions. Properties of proposed estimator have been studied after estimating the constants involved in the estimator and hence a modified regression type estimator has been suggested which has superiority over Isaki (1983) type regression estimator of population variance. At last numerical illustration have been made.

KEYWORDS: Auxiliary variable, Ratio and Regression type estimators, Bias, Mean square error (MSE), Relative Efficiency, Simple Random Sampling.

1. INTRODUCTION:

The problem of estimating population variance arises in many practical situations like genetical, biological and medical studies Bland and Altman (1999), Singh et al (1988). The problem has been well dealt in literature in simple random sampling. Wakimoto (1971) considered this problem in stratified sampling while Tripathi (1977), Liu (1974), Chaudhury (1978), Mukhopadhyay (1978), Swain and Mishra (1994) have paid their attention in PPS and general sampling designs. Padmawar and Mukhopadyay (1981). Mukhopadhyay (1982) made a significant contribution for estimating population variance under super-population models. Chaudhury and Adhikari (1990) made an attempt for its estimation, using randomized response technique. Dutta and Ghosh (1993) extended their view under Bayesian Approach. Taking advantage of high correlation between study and auxiliary variables, Isaki (1983) proposed ratio and regression type estimators of population variance. Birader and Singh (1988), Agrawal and Panda (1999) explored their discussion under prediction approach.

Assume that the finite population consists of N identifiable units (U_1, U_2, U_3, ………., U_N ), taking the values ( y_1, y_2, y_{3,……….,}y_N ) on study variable y and ( x_1, x_2, x_{3,……….,}x_N) on auxiliary variable. Let be population variance of y, where . Let a sample s = (1,2,…j,..,n) be taken from the population U by simple random sampling procedure. Let (y_j, x_j), j = 1,2,…,n be the values of (y,x) on s.

Then it is well known in literature that sample variance is an unbiased estimator of (or ), if units are taken by with (or without) replacement procedure. Singh et al (1973), Searls and Intarapanich (1990) proposed an estimator

Where K₁ is a suitably constant which depends upon the coefficient of kurtosis of y. The estimator is found to be more efficient than, if sample size n is small. Replacing by ratio type estimator in (1.1). Prasad and Singh (1990) proposed modified ratio estimator of population variance which is almost equally efficient as for large samples.

Utilizing information on auxiliary variable x, Das and Tripathi (1978) proposed some estimators of population variance if population mean, variance or coefficient of variation of x is known. Srivastava and Jhajj (1980) proposed a class of estimators of under the knowledge of and studied its properties under certain regularity conditions. This estimator is equally efficient as difference type estimator.

which is found to be considerably more précised than if correlation between x and y is high. The quest for improvement over lead Singh et al (1988) to propose the following estimator of as

which is marginally more efficient than for small samples. Noting that sum of coefficients of, and in is unity, Dubey and Kant (2001) proposed estimator of as

where K₄ and K₅ are suitable constants. The estimator is more efficient than all the above estimators if is closure to.

In section 2, a generalized estimator has been proposed which is more efficient than all the above estimators under very obvious conditions.

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