An Alternative Ratio-cum-product Estimator of Finite Population Mean Using Coefficient of Kurtosis of Two Auxiliary Variates in Two-phase Sampling

This paper deals with the problem of estimation of population mean in two-phase sampling. A ratio-product estimator of population mean using known coefficient of kurtosis of two auxiliary variates has been proposed. In fact, it is a two-phase sampling version of Tailor et al. (2010) estimator and its properties are studied. Proposed estimator has been compared with usual unbiased estimator, classical ratio and product estimator in two-phase sampling, and two-phase sampling versions of Singh (1967) and Singh et al. (2004) estimators respectively. To judge the merits of the proposed estimator over other estimators an empirical study is also carried out.


Introduction
Auxiliary information plays a very important role in improving the efficiencies of estimator(s) of population parameter(s).Ratio, product and regression methods are good examples in this context.These methods require knowledge of population mean of auxiliary.Use of coefficient of kurtosis of auxiliary variate has also been in practice for improving the efficiency of the estimators of finite population mean.In some practical situations population coefficient of variation and coefficient of kurtosis of auxiliary variate x are known. [1]Ajagaonkar (1975) and [12] Sisodia and Dwivedi (1982) discussed double sampling procedure using single auxiliary variate whereas [3] Khan and Tripathi (1967), [6] Rao (1975) and [7] Singh and Namjoshi (1988) considered the use of multiauxiliary variates in double sampling. [10]Singh (1967) used information on two auxiliary variates and defined a ratio-product estimator assuming that population mean of auxiliary variates are known.In the line of [11] Sisodia and Dwivedi (1981) [9] Singh et al. (2004) proposed a ratio type estimator using coefficient of kurtosis. [18]Tailor et al. (2010) suggested a ratio-cum-product estimator using coefficient of kurtosis of two auxiliary variates in simple random sampling while [16] Tailor and Sharma (2013) suggested a ratio-cum-product estimator using double sampling.In this research paper authors have suggested [18]  .Let the auxiliary variables 1 x and 2 x be positively and negatively correlated with the study variate y respectively.
Let us suppose that / be the population mean of the study variate y and auxiliary variates ) , ( be the unbiased estimators of population mean Y , 1 X and 2 X respectively.
The classical ratio and product estimators for estimating population mean Y are respectively defined by ( Assuming that population means 1 X and 2 X of the auxiliary variables 1 x and 2 x are known, [9] , (1.4) [10] Singh (1967) suggested a ratio-product estimator using information on two auxiliary variates 1 x and 2 x to estimate population mean Y as The problem of estimating population mean Y of y when the population means 1 X and 2 X of 1 x and 2 x are known, has been dealt at a great length in the literature see [8] Singh and Tailor (2005), [17] Tailor and Tailor (2008) and many others.However, in many practical situations when no information is available on the population means 1 X and 2 X of 1 x and 2 x in advance before starting the survey, we estimate Y from a sample obtained through a two phase selection.Adopting simple random sampling without replacement (SRSWOR) scheme at each phase, the two-phases (or double) sampling scheme is as follows: i.A first phase sample 1 S of fixed size n' is drawn form U to observe only 1 x and 2 x to estimate 1 X and 2 X respectively. ii.
A second phase sample 2 S of fixed size n is drawn from 1 S to observe y only or second phase sample may be drawn independently to the first sample i.e. two cases, designated as-

Case I :
As a sub sample from the first phase sample, Case II : Draw independently to the first phase sample.
In two-phase or double sampling, the usual ratio and product estimators of population mean Y are respectively defined as where y , 1 x and 2 x are sample means based on second phase sample of size n whereas are the first phase sample means of 1 x and 2 x , which are unbiased estimates of population means 1 X and 2 X respectively of auxiliary variate x.
Two-phase sampling versions of [9] Singh et al. (2004) ratio and product type estimators of population mean Y are defined by Two-phase sampling version of [10] Singh (1967) ratio-product estimator of population mean Y is defined by (1.10) We obtain the bias and mean squared error of two-phase sampling versions of estimators considered in this section to the first degree of approximation.denote the bias and the mean squared error under case I and II respectively which are given as It is well known under Simple random sampling without replacing (SRSWOR) variance of unbiased estimator is defined as 2. Proposed Ratio-Product Estimator [18] Tailor et al. (2010) proposed ratio-product estimator of population mean Y using information on coefficient of kurtosis x x   of auxiliary variates 1 x and 2 x as The estimator T Y ˆ requires the knowledge of 1 X and 2 X .When information is not available, we define T Y ˆ in two-phase sampling as To obtain the bias and mean squared error of  ) is more efficient than the usual unbiased estimator y if

Efficiency Comparisons of
It is noted from (1.21) and (2.6) that the suggested estimator is more efficient than usual two-phase sampling ratio estimator Comparison of (1.22) and (2.6) shows that the suggested estimator would be more efficient than usual two-phase sampling product estimator Comparing (1.23) and (2.6) reveals that the suggested estimator is more efficient than the two-phase sampling version of Singh et al. ( 2004) ratio type estimator From (1.24) and (2.6) it is observed that the suggested estimator is more efficient than the two-phase sampling version of Singh et al. ( 2004) product type estimator Comparison of (1.25) and (2.6) that the suggested estimator is more efficient than the two-phase sampling version of Singh (1967) ratio-cum-product type estimator Expressions (3.1) to (3.6) are conditions for case I under which suggested estimator has less mean squared error than usual unbiased estimator y , usual two-phase sampling ratio estimator ) d ( R y , two-phase sampling product estimators ) d ( P y , two-phase sampling versions of [9]  respectively and the two-phase sampling version of [10] Singh (1967) estimator  ), would be more efficient than (ii) two-phase sampling ratio estimator (iii) two-phase sampling product estimator (iv) Singh et al's (2004) two-phase sampling ratio type estimator Singh et al's (2004) two-phase sampling product type estimator (vi) Singh (1967) two-phase sampling ratio-cum-product type estimator y , two-phase sampling versions of estimators suggested by [9]  Y ˆ) and [10] Singh (1967) two-phase sampling ratio-product type estimator

Empirical Study
To analyze the performance of the proposed estimator of population mean Y in two- phase sampling in comparison to other estimators, one natural population data set is being considered.We have computed Percent relative efficiencies (PREs) of 26), (1.27), (1.28), (1.29), (1.30), (1.31) and (2.7) it is observed that the suggested

1 x 2 x
y .The description of the population is given below.Population: [Source: Steel and Torrie (1960, p.282)] y = Log of leaf burn in seconds,  Potassium percentage,  Chlorine

(
I and II] are more efficient than the usual unbiased estimator y , ratio estimator in two- phase sampling ) (d R y , two-phase sampling product estimator ) (d P y , two-phase sampling versions of Singh et al. (2004) ratio and product type estimators two-phase sampling version of Singh (1967) estimator.Larger gain in efficiency is observed by using proposed estimators over other estimators.It is also observed that When samples taken as a sub sample from the first phase sample) is giving better result as compare to for use in practice.

Table 5 .
1 and 5.2 exhibits that the suggested estimators