A General Class of Dual to Ratio Estimators

In this paper we have considered the problem of estimating the population mean using auxiliary information in sample surveys. A class of dual to ratio estimators has been defined. Exact expressions for bias and mean squared error of the suggested class of dual to ratio estimators have been obtained. In particular, properties of some members of the proposed class of dual to ratio estimators have been discussed. It has been shown that the proposed class of estimators is more efficient than the sample mean, ratio estimator, dual to ratio estimator and some members of the suggested class of estimators in some realistic conditions. Some numerical illustrations are given in support of the present study.


Introduction
The use of ratio method of estimation is quite effective if the correlation between the study variable y and the auxiliary variable x is positive (high). On the other hand, if this correlation is negative (high) product method of estimation is employed for estimating population mean Y of the study variable y. It is to be mentioned that the ratio estimator suffers with a drawback that it does not provides exact bias and mean squared error while the product method of estimation provide the exact bias and mean squared error. But in practice positive correlation between the two variables   x y, are generally encountered, while negative correlation situation is not much as compared to positive correlation. Srivenkataramana (1980) and Bandyopadhyay (1980) have advocated the use of product method of estimation in case of positive correlation using simple transformation which induce the negative correlation even if the correlation between the two variables   respectively. Suppose that a simple random sample of size n is drawn without replacement from U for estimating the population mean Y of the study variable y. Let   x y, be the sample means of   x y, respectively based on n observations drawn from the population U. Using the transformation: is the mean of the unobserved units in the population U and X is the known population mean of the auxiliary variable x. The exact bias of the estimator SB y is given by For exact mean squared error of the estimator SB y the reader is referred to Srivenkataramana (1980). To the first degree of approximation, the mean squared error (MSE) of the estimator SB y is given by Under simple random sampling without replacement (SRSWOR) the variance/MSE of the usual unbiased estimator y of population mean Y is given by For estimating the population mean Y , when the correlation between the two variables   x y, is positive and population mean X of the auxiliary variable x is known, the classical ratio estimator is defined by To the first degree of approximation, the bias and MSE of the ratio estimator R y are respectively given by In the present paper we have suggested a generalized version of the dual to ratio estimator SB y along with its properties. Numerical illustration is given in support of the present study.

SB y
Keeping in view the form of the dual to ratio estimator SB y at (1.2), we define its generalized version for population mean Y as where 'b' is suitable chosen scalar such that which is due to Srivenkataramana (1980) and Bandyopadhyay (1980).
Many more acceptable estimators can be generated from the proposed estimator The biases and mean squared errors of the estimators belonging to the suggested estimator H y can be easily obtained from (2.6), (2.7) and (2.8) just by putting different values of the scalar 'b'.

Efficiency Comparison
In this section we have obtained the regions of preferences in which the suggested estimator H y is better than the usual unbiased estimator y , ratio estimator R y and dual to ratio estimator or equivalently, Thus we state the following theorem.
(3.9)   and yields an estimator more efficient than R y or y .

Allowable Departure from Optimum
The optimum value for population mean Y . Writing (5.2) in terms of 0 e and 1 e we have  Thus, it follows from (5.6) that to ensure only a small relative increase in MSE,  must be close to zero if  is high but can depart substantially from zero if  is just moderate.
A General Class of Dual to Ratio Estimators 428

Estimator Based on Estimated Optimum
It is observed from (4.2) that the asymptotically optimum estimator (AOE) HO y presupposes the knowledge about k. If the value of k cannot be guessed quite accurately, then the only alternative left to the investigator is to replace k in (4.2) by its consistent estimator kˆ obtained from the data in hand. Thus the estimator of the population mean, Y of y on the estimated optimum is is a consistent estimate of  Srivenkataramana (1980) and Bandyopadhyay (1980) can be given by (6.5) where b is same as defined earlier.
For b=0, * H y reduces to usual unbiased estimator y while for f b  , it reduces to Srivenkataramana (1980) and Bandyopadhyaya (1980) dual to product estimator defined by

Empirical Study
To judge the merits of the suggested estimator over usual unbiased estimator y , ratio estimator R y , dual to ratio estimator SB y , we have considered three natural population data sets whose descriptions are given below.