AN ALTERNATIVE RATIO-CUM-PRODUCT ESTIMATOR OF POPULATION MEAN USING A COEFFICIENT OF KURTOSIS FOR TWO AUXILIARY VARIATES

An alternative ratio-cum-product estimator of population mean using the coefficient of kurtosis for two auxiliary variates has been proposed. The proposed estimator has been compared with a simple mean estimator, the usual ratio estimator, a product estimator, and estimators proposed by Singh (1967) and Singh et al. (2004). An empirical study is also carried out in support of the theoretical findings.


INTRODUCTION
The use of known parameters for auxiliary variates has played an important role in improving the efficiencies of estimators.Sisodiya and Dwivedi (1981) used the coefficient of variation for auxiliary variates.Later, Singh et al. (2004) used the coefficient of kurtosis for auxiliary variates.Upadhyaya and Singh (1999) derived ratio and product type estimators using both a coefficient of variation and a coefficient of kurtosis for auxiliary variates.Singh (1967) utilized information on two auxiliary variates 1 x and 2 x and suggested a ratio-cum-product estimator for population mean.This paper is an attempt to study the use of a coefficient of kurtosis (

=
be a finite population of N units.Suppose two auxiliary variates 1 x and 2 x are observed, along with study variate y, on , where 1 x is positively and 2 x is negatively correlated with y.A simple random sample of size n is drawn by simple random sampling without replacement (SRSWOR) from the population U to estimate the population mean ( Y ) of study character y when the population means x and 2 x , respectively, are known.
The usual ratio and product estimators for estimating the population mean Y are given respectively by Singh et al. (2004) defined a ratio and product type estimator using the coefficient of kurtosis ( To estimate Y , Singh (1967) suggested a ratio-cum-product estimator using information on two auxiliary variates 1 x and 2 x as To the first degree of approximation, the mean squared error (MSE) of the estimators R y , P y , RB y , PB y , and 1 Ŷ are given respectively by [ ] )

PROPOSED ESTIMATOR
Assuming that the information on the coefficient of kurtosis ( ) for auxiliary variates 1 When information on the second auxiliary variable 2 x is not available (or equivalently, the variable 2 x takes only a constant value, i.e., i x 2 = a (constant); i = 1,2,…,N), the estimator 2 Ŷ reduces to RB y as suggested by Singh et al.   (2004).On the other hand, if the information on the auxiliary variable 1 x is not available (or equivalently the Data Science Journal, Volume 9, 24 July 2010 variable 1 x takes only a constant value, i.e., i x 1 = a* (constant); i =1,2,…,N), the estimator turns out to be the estimator PB y , a product version of RB y .
To obtain the bias and mean squared error of the proposed estimators, we assume that 0 (1 , and The bias and mean squared error of 2

EFFICIENCY COMPARISON
The variance of sample mean y in simple random sampling without replacement (SRSWOR) is From equations ( 6) to ( 9), ( 14), and (15) we have .

A FAMILY OF UNBIASED ESTIMATORS USING THE JACKKNIFE TECHNIQUE
Suppose a simple random sample of size gm n = is drawn without replacement and split at random into g sub-samples, each of size m.Then the Jack-knife type ratio-cum-product estimator for population mean Y , using 2 Ŷ is given as ,2 are the sample means based on a sample of (n-m) units obtained by omitting the th j group and j y and ij x (i=1,2; j=1,2,…,g) are the sample means based on the th j sub samples of size m=n/g.
The bias of J Y 2 ˆ, up to the first degree of the approximation bias of From ( 13) and ( 21) we have Upon simplifying (22), we get a general family of almost unbiased ratio-cum-product estimators of Y as (24) This is the Jack-knifed version of the proposed estimator 2 Ŷ .

AN OPTIMUM ESTIMATOR IN FAMILY
where ) ( 2 Y MSE is given by (14).
Using ( 14), (15), and (30) in ( 26), the variance of u Y 2 ˆ up to the terms of order 1 n − is given as which is minimized for The optimum value * opt b of * b can be obtained quite accurately through past data or experience.