A Class of Estimators of Population Mean in Case of Post Stratification

This paper proposes a class of ratio-cum-product type estimators in case of post-stratification. Particular members of the proposed class of ratio-cum-product type estimators have been identified and studied thoroughly from efficiency point of view. It has been shown that the identified particular estimators are more efficient than the usual unbiased estimator, Ige and Tripathi (1989) estimators, Chouhan (2012) estimators, Tailor et al. (2016) estimator and other considered estimators. An empirical study has been carried out to demonstrate the performance of the proposed estimators.


Introduction
and Robson (1957) envisaged classical ratio and product estimators which were studied in case of post stratification by Ige and Tripathi (1989).Recently, Lone and Tailor (2014) and Lone and Tailor (2015) proposed ratio and product type exponential estimators in case of post-stratification.Chouhan (2012) proposed class of ratio type estimators using various known parameters of auxiliary variates in case of post stratification.Tailor et al. (2015) proposed dual to Ige and Tripathi (1989) ratio and product estimators.Tailor et al. (2016) proposed a ratio-cum-product type estimator in case of post-stratification.Singh (1967) used information on population mean of two auxiliary variates and proposed ratio-cum-product type estimator for population mean in simple random sampling.Singh (1967) and Chouhan (2012) motivate authors to propose the class of ratio-cum-product type estimators in case of post-stratification.Many Researchers including Holt and Smith (1979), Jagers et al. (1985), Jagers (1986), Ige and Tripathi (1989), Agrawal and Panday (1993) : th h stratum mean for the auxiliary variate x , : th h stratum mean for the study variate , y : th h stratum mean for the auxiliary variate z , : Population mean of the study variate y and Population mean of the auxiliary variate z .
A sample of size n is drawn from population U using simple random sampling without replacement.After selecting the sample, it is observed that which units belong to th h stratum.Let h n be the size of the sample falling in th h stratum such that Here it is assumed that n is so large that possibility of h n being zero is very small.In case of post-stratification, usual unbiased estimator of population mean Y is defined as where is the weight of the Using the results from Stephen (1945), the variance of PS y to the first degree of approximation is obtained as where Ige and Tripathi (1989) defined a ratio and a product type estimator in case of poststratification as ( where Chouhan (2012) proposed the following ratio type estimators for population mean Y in case of post-stratification as Where as the product version of Upto the first degree of approximation, mean squared error of the estimators where 2016) proposed a ratio-cum-product type estimator in case of poststratification as Up to the first degree of approximation, mean squared error of RP PS Y ˆ is obtained as (1.20)

Proposed Class of Ratio-Cum-Product Type Estimators
In the line of Singh (1967), we suggest a class of ratio-cum-product type estimators for estimating population mean Y as To obtain the bias and mean squared error of the proposed estimator t , we write Expressing (2.1) in terms of e's, we have where Taking expectation on both sides of (2.2), we get the bias of the proposed estimator t upto the first degree of approximation as Squaring and taking expectation on both sides of (2.2), the mean squared error of the proposed estimator t , upto the first degree of approximation is obtained as Equation (2.4) can also be written as The MSE of t is minimum when Putting (2.6) in (2.5), we get the minimum mean squared error of the proposed estimator

Empirical Study
To exhibit the performance of the proposed estimators, two population data sets are being considered.Descriptions of data sets are given by

Conclusion
A class of ratio-cum-product type estimators for population mean has been defined.The usual unbiased estimator, Ige and Tripathi (1989) estimators, Chouhan (2012) estimators and Tailor et al. (2016) estimator have been identified to be the member of the proposed class of ratio-cum-product type estimators t .Section 3 provides the conditions under which the proposed estimator t has less mean squared error as compared to the mean squared error of the other considered estimators.It is observed that particular members t are recommended for use in practice.Thus, it has been concluded that there is an enough scope of generating better estimators from the proposed class of ratio-cumproduct type estimator t using suitable known parameters of auxiliary variates.
Singh and Ruiz Espejo (2003) Tailor et al. (2011) and Tailor et al. (2015) contributed well in case of post-stratification.Let us consider a finite population y be the study variate and x is the auxiliary variate correlated positively with the study variate y and z is the auxiliary variate, negatively correlated with the study variate y .Let hi y be the observation on auxiliary variates x and z respectively , then function of population parameters of the auxiliary variates.

2016 pp111-124 121 4. Study on the particular members of the proposed class of ratio-cum-product type estimators t
To illustrate the general result, we have considered the new members Pak.j.stat.oper.res.Vol.XII No.1