Ratio Estimators in Simple Random Sampling Using Information on Auxiliary Attribute

Some ratio estimators for estimating the population mean of the variable under study, which make use of information regarding the population proportion possessing certain attribute, are proposed. Under simple random sampling without replacement (SRSWOR) scheme, the expressions of bias and mean-squared error (MSE) up to the first order of approximation are derived. The results obtained have been illustrated numerically by taking some empirical population considered in the literature.


Introduction
The use of auxiliary information can increase the precision of an estimator when study variable y is highly correlated with auxiliary variable x.There exist situations when information is available in the form of attribute φ , which is highly correlated with y.For example a) Sex and height of the persons, b) Amount of milk produced and a particular breed of the cow, c) Amount of yield of wheat crop and a particular variety of wheat etc.
Consider a sample of size n drawn by SRSWOR from a population of size N. Let i y and i φ denote the observations on variable y and φ respectively for i th unit ) N ,.... 2 , 1 i ( = .Suppose there is a complete dichotomy in the population with respect to the presence or absence of an attribute, say φ , and it is assumed that attribute φ takes only the two values 0 and 1 according as Taking into consideration the point biserial correlation between a variable and an attribute, Naik and Gupta [2] defined ratio estimator of population mean when the prior information of population proportion of units, possessing the same attribute is available, as follows: Here y is the sample mean of variable of interest.The MSE of NG t up to the first order of approximation is - Where In the present paper, some ratio estimators for estimating the population mean of the variable under study, which make use of information regarding the population proportion possessing certain attribute, are proposed.The expressions of bias and MSE have been obtained.The numerical illustrations have also been done by taking some empirical populations considered in the literature.

The suggested estimator
Following Ray and Singh [3], we propose the following estimator - .
Remark 1: When we put 0 b = φ in (2.1) the proposed estimator turns to the Naik and Gupta [2] ratio estimator NG t given in (1.1).
MSE of this estimator can be found by using Taylor series expansion given by - Expression (2.2) can be applied to the proposed estimator in order to obtain MSE equation as follows: ) Several authors have used prior value of certain population parameters (s) to find more precise estimates.Searls (1964) used Coefficient of Variation (CV) of study character at estimation stage.In practice this CV is seldom known.Motivated by Searls (1964) work, Sen (1978), Sisodiya and Dwivedi (1981), and Upadhyaya and Singh (1984) used the known CV of the auxiliary character for estimating population mean of a study character in ratio method of estimation.The use of prior value of Coefficient of Kurtosis in estimating the population variance of study character y was first made by Singh et. al. (1973).Later, used by and Searls and Intarapanich (1990), Upadhyaya and Singh (1999), Singh (2003) and Singh et. al. (2004) in the estimation of population mean of study character.Recently Singh and Tailor (2003) proposed a modified ratio estimator by using the known value of correlation coefficient.
In next section, we propose some ratio estimators for estimating the population mean of the variable under study using known parameters of the attribute φ such as coefficient of variation C p , Kurtosis ( ) and point biserial correlation coefficient pb ρ .

Suggested Estimators
We suggest following estimator -

Where
) 0 ( m 1 ≠ , m 2 are either real number or the functions of the known parameters of the attribute such as C p , ( ) and pb ρ .
The following scheme presents some of the important estimators of the population mean, which can be obtained by suitable choice of constants m 1 and m 2 : Following the approach of section 2, we obtain the MSE expression for these proposed estimators as

Efficiency comparisons
It is well known that under simple random sampling without replacement (SRSWOR) the variance of the sample mean is

i φ = 1 ,
if ith unit of the population possesses attribute φ = 0, otherwise.in the population and sample respectively possessing attribute φ .
4), we get MSE of t 1 as Naik and Gupta (1996)s satisfied, proposed estimators are more efficient than the sample mean.Now, we compare the MSE of the proposed estimators with the MSE ofNaik and Gupta (1996)estimator NG t .From (3.2) and (1.1) we have