Some Improved Estimators in Double Sampling Using two Auxiliary Variables

Dash and Mishra [1] suggested an improved class of estimators without defining the optimum estimator. However, they gave the wrong Taylor’s series expression of their class of estimator and their minimum mean squared error expressions are also incorrect. Here we show that Ahmed et al.’s [2] class of chain estimators is more efficient than Dash and Mishra’s [1], with minimum mean squared error.


Introduction
uppose one is interested in estimating the population mean of a study variable Y from a finite population of size N. If the information on an auxiliary variable X which is highly correlated with Y is readily available on all the units of the population, it is well known that ratio and regression type estimators can be used for increased efficiency, incorporating the knowledge of ̅ . However, in certain practical situations ̅ is not known a priori, in which case the technique of Double sampling is fruitfully applied. The values of X are assumed to be known over a large simple random sample of size . Now suppose that information on yet another auxiliary variable Z is available on all units of the population ( ̅ is the population mean of Z). Then we observe the study variable Y from a simple random subsample of that large sample of size ( ) to estimate the population mean ( ̅ ) of study variable . The first phase sample of size ( ) is selected to observe and , and the second phase sample of size is selected to observe and . Suppose ̅ and ̅ are the unbiased estimators of ̅ and ̅ , respectively, based on first phase sample ; ̅, ̅ and ̅ are the unbiased estimators of ̅ , ̅ and ̅ , respectively, based on second phase sample, . Thus, a specific class of estimators can be suggested by suitably choosing the auxiliary information. Over a period, a large number of estimators were proposed under different assumptions which can be classified as a particular member or case of some general class of estimators. In section 2, we critically evaluate some of these estimators.

Some Classes of Estimators
Using information on two closely related auxiliary variables, X and Z, a good number of classes of improved estimators were suggested by Chand [3], Kiregyera [4,5], Srivastava et al. [6], Sahoo et al. [7], Tripathi and Ahmed [8], and Mishra and Rout [9]. A brief review of these estimators may be referred to Das and Mishra [1].
Dash and Mishra [1] suggested an improved class of estimators as defined Ahmed [10] proposed a wider class of estimators as is a function such that ̅ ̅ satisfying some regularity conditions.
Tripathi and Ahmed [8] suggested the general class of estimator: where and are suitably chosen statistics such that or .
The optimum values of , and are respectively , and , where and are partial regression coefficients and is a the total regression coefficient (See Tripathi and Ahmed [8]).

The optimum estimator (1.3) is
where and are sample partial, and is the sample total, regression coefficients respectively.
Ahmed et al. [2] suggested another general class of estimators: where , and are suitable chosen constants.
The optimum values of , and are respectively, It can be shown that the suggested estimators of Chand [3], Srivastava et al. [6], Samiuddin and Hanif [14] and Hanif et al. [15] are the particular cases of the class of estimators (1.5) and that it has minimum mean squared error (MSE).
The minimum MSE of (1.2), (1.3) and (1.5) is the same and it is where is the population correlation coefficient between y and z, and is the multiple correlation coefficient of y on x and z ( √ . It can be easily shown that the proposed improved class of estimators of Dash and Mishra [1] as given in (1.1) does not accommodate the estimator (1.5), as the known mean ̅ does not contain in (1.1).
They were not able to provide a correct Taylor series expression of the estimator (1.1), which is a parametric function. Besides this, they failed to present an explicit estimator for population mean. In their article, Dash and Mishra [1] presented an incorrect expression of the minimum variance of Sahoo et al.'s [7]

Conclusion
This paper provides an overview of classes of estimators in Double sampling using two auxiliary variables. Some limitations and errors of an improved class of estimators recently suggested by Dash and Mishra [1] have been pointed out. The reviews suggest that although a lot of improved classes of estimators have been presented in recent time, they are equivalent or particular cases of the general class of estimators suggested by Tripathi and Ahmed [8] and Ahmed et al. [2]. Diana and Tommassic [18] argued that the regression type of estimators as given by Ahmed et al. [2] is one of the best estimators, at least at the first order of approximation. They further suggested that no new estimators be proposed, since no estimator can be more efficient than the best estimators proposed by Ahmed et al. [2], and some others, at least at the first order of approximation.