Efficient Ratio-type Exponential Estimator for Population Variance

Received: 02/Jun/2018, Revised: 09/May/2018, Accepted: 23/Jun/2018, Online: 30/Jun/2018 Abstract-In this paper, a new exponential ratio type estimator has been proposed for estimating the population variance using auxiliary information. To the first order of approximation, i.e., to o(n -1 ), the expressions for the bias and the mean square error of the proposed exponential ratio-type estimator have been derived. The optimum value of the characterizing scalar, which minimizes the MSE of proposed estimator, has been obtained. With this optimum value, the expression for minimum MSE of the proposed estimator has been arrived at. The proposed estimator has been compared theoretically with sample variance, traditional ratio estimator due to Isaki [1], and exponential ratiotype estimator due to Singh et.al.[3] and it is found that, under practical conditions, the proposed estimator fares better than its competing estimators. An empirical investigation has been carried out to demonstrate the efficiency of the proposed estimator.


INTRODUCTION
In survey sampling, the utilization of auxiliary information is frequently acknowledged to enhance the accuracy of the estimation of population characteristics. Estimation of the finite population variance has great significance in various fields such as in matters of health, variation in body temperature, pulse beat and blood pressure etc. Using auxiliary information, we, in this paper, introduce one new estimator which fares better than competing estimators.
Consider a finite population of size N, arbitrarily labelled 1, 2....N. Let and be, respectively, the values of the study variable y and the auxiliary variable x, in respect of the i th unit (i=1, 2,… N) of the population. When the auxiliary variable x is positively correlated with the study variable y and , the population variance of x is known, ratio method of estimation is usually invoked to estimate the population variance of the study variable.

II. NOTATIONS AND SOME EXISTING ESTIMATORS
In simple random sampling without replacement, we know that the sample variance provides an unbiased estimator of the population variance , Accordingly, we define Isaki (1983) proposed the ratio type estimator for estimating the population variance of the study variable as = (2) whose bias and mean square error, up to the first order of approximation ,i.e., to o ( ) are respectively, and

III. PROPOSED RATIO-TYPE EXPONENTIAL ESTIMATOR
We propose a new ratio-type exponential estimator for estimating the population variance , which is given by Retaining only up to 2 nd term, we get The bias of the proposed exponential ratio estimator, to the first degree of approximation, i.e., to o ( ), The mean square error of the proposed exponential ratio estimator, to the first degree of approximation, i.e., to o ( ), has been derived as follows With a view to determining the most suitable value of , to be called , we proceed to minimize the mean square error subject to variation in , implying thereby that On comparison of (13) with (4), the following results can be arrived at On comparison of (13) with (7), the following results can be arrived at ˂ .
(1)The newly proposed estimator performs better than the simple variance estimator of population variance if .
(2)The newly proposed estimator performs better than the ratio-type estimator due to Isaki (1983) for variance if .
(3)The newly proposed estimator performs better than the ratio-type exponential estimator due to Singh et. al.

IV. EMPRICAL FINDINGS
With a view to establishing the supremacy of the proposed estimator over the competing estimator, we consider the following example for ratio method of estimation which is taken from Sukhatme and Sukhatme (  Thus, we find that the condition (16) is satisfied.
The MSEs of the competing estimators have been computed and presented in Table 4.2 The percentage relative efficiency of the proposed estimator, over the competing estimator , and has been given in the table 4.3 The percentage relative efficiency (PRE) of different estimators with respect to usual unbiased estimator is computed by the formula PRE (., ) = ⤫100.
It is clear from the above table that the newly proposed estimator performs better than the competing estimators.

V. CONCLUSION
We have proposed a ratio-type exponential estimator for estimating the population variance and demonstrated both theoretically and numerically that the proposed estimator fares better than its competing estimators under conditions that hold good in practice. This work can be extended to two-phase sampling also.