A Two Parameter Ratio-Product-Ratio Estimator in Post Stratification

In this paper we consider a two parameter ratio-product-ratio estimator for estimating population mean in case of post stratification following the estimator due to Chami et al (2012). The bias and mean squared error of proposed estimator are obtained to the first degree of approximation. We derive conditions under which the proposed estimator has smaller mean squared error than the sample mean ps y , ratio estimator ) ( ps R y and product estimators ) ( ps P y . Empirical studies gives insight on the magnitude of the efficiency of the estimator developed.


Introduction:
The principal aim of statistical surveys is to obtain information about of interest.To increase the precision of the estimates we use information on the auxiliary variable.Stratification is one of the most widely used techniques, requires the size of the strata as well as sampling frame for each stratum.But in many situations sampling frame is not available.For example in households surveys it is possible to know the number of families added in a locality but which families belong to which locality up to date sampling frame of different strata may not be possible.In this type of situation, post stratification technique is used.The post stratification was first introduced by Holt and Smith (1979) and Ige and Tripathi (1989).It is known that when the auxiliary information is used at the estimation stage, the ratio estimator is best among a wide class of estimators when the relation between y and x, the variate under study and auxiliary variate, respectively is a straight line through the origin and the variance of y about this line is proportional to x.In such a situation the ratio estimator is as good as regression estimator.In many practical situations, the regression line does not pass well as that of regression estimator.Keeping this fact in view and also due to the stronger intuitive appeal, statisticians are more inclined towards the use of the ratio and the product estimators and hence a large amount of work has been carried out towards the modification of ratio and product estimators, for instance, see Singh (1986), Singh and Espejo (2003) etc.. Motivated by this Ige and Tripathi have suggested an improved version of combined/separate ratio and product estimators in post stratified sampling and studied their properties under large sample approximation.Ige and Tripathi (1989) have conducted an empirical study in support of their studies.Tuteja et al (1995) have extended the study of Ige and Tripathi (1989) based on post stratification and auxiliary information.Recently Chouhan (2012) proposed class of ratio type estimators using various known parameters of auxiliary variates in case of post stratification.Keeping in this view we have suggested a two parameter ratio-product-ratio estimator for estimating population mean in case of post stratification adapting the estimator due to Chami et al (2012).For a finite population of size N, we are interested in estimating the population mean Y of main variable y.Use of auxiliary information has been in practice for improving the efficiency of estimator(s).Usually, auxiliary information is easily available with study variate with little extra cost and efforts.Let us consider a finite population ( ) . A sample of size n is drawn from population U using simple random sampling without replacement (SRSWOR).After selecting the sample, it is observed that which units belong to h th stratum.Let nh be the size of the sample falling in h th stratum such that Here it is assumed that n is so large that the probability of nh being zero is very small.Let hi y be the observation on i th unit that fall in h th stratum for study variate y and hi x be the observation on i th unit that fall in nh stratum for auxiliary variate x, then, is sample mean of h n sample units that falls in the th h stratum.
We denote the population variances/mean squares of Y and X as ( )

(
), where .X Y R = Following Khoshnevisan et al (2007), Onyeka (2012) proposed a class of estimators for population mean Y in case of post stratified sampling as where ( )  , are real constants and ( ) , b are either constants or functions of known population parameters of the auxiliary variates such as standard deviation ( )  and correlation coefficient yx  between y and x.
To the first degree of approximation, the MSE of pss y is given by It is to be mentioned that Onyeka (2012) generated five new ratio-type estimators forY : A Two Parameter Ratio-Product-Ratio Estimator in Post Stratification 276 respectively and eight product-type estimators for )  ( ) , , Motivated by Bhal and Tuteja (1991), Tailor et al (2017) suggested the following combined ratio-type and producttype exponential estimators for population mean Y of y in post stratified sampling as and To the first degree of approximation, the MSEs of ps Y Re ˆand ps Pe Y ˆare respectively given by Bacanli and Aksu (2012) proposed the following separate ratio estimators for the population mean Y of y in post stratified sampling as , ) In this paper we have developed a two parameter ratio-product-ratio estimator in post stratification on the line of Chami et al (2012).Combined as well as separate estimators are proposed and their properties are studied under large sample approximation.Numerical examples are given in support of the present study.

Proposed two parameter combined ratio-product-ratio type estimator
Taking motivation from Holt and Smith (1979), Ige and Tripathi (1989) and Chami et al. (2012) for estimating the population mean Y in case of post stratification we suggest the following two parameter ratio-product-ratio estimator where ( ) (ii)  and to the first degree of approximation (i.e. up to order n -1 ); ( )


Now expressing (2.1) in terms of ei's, we have We assume that , so that ( ) Now expanding the right hand side of 2) and neglecting terms of ei's having power greater than two we have Taking expectations of both sides of (2.3), we get the bias of to the first degree of approximation as The suggested ratio-product-ratio estimator • Mean Squared Error of Squaring both sides of (2.3) and neglecting terms of ei's having power greater than two, we have Taking expectations of both sides of (2.6) we get the mean squared error of to the first degree of approximation as Taking the gradient Inserting (2.8) to zero to get the critical points, we obtain the following solutions One can see that the critical point in (2.9) is a saddle point unless C=0, in which we get a local minimum.However, the critical points obtained by (2.10) are always minima, for a given C, (2.10) is the equation of hyperbola symmetric through ( ) ) to the first degree of approximation is given by where

Comparison of MSEs and Selection of Parameters
In this section we present the comparison of the proposed estimator It is well known that when (i Thus the proposed ratio-product-ratio estimator is more efficient than usual unbiased estimator ps y as long as the conditions (i)-(iv) are satisfied.From (1.4) and (2.7) we have

Comparing the MSE of the Ratio Estimator
Hence from solution (i), where C>1, we have the following , we obtain the following

C
(if both the factors in (3.4) are negative).
Hence from solution (i), where C >1, we have the following , we obtain the following We mention that this implies 2 , and the range for  and  , where these inequalities hold are explicitly given by the following two situations: In situation (ii), where , the following range of  and  can be obtained.

Comparison of the Proposed Class of Estimators
or equivalently, Let ( ) be pre-assigned.Then the proposed class of estimators Thus for given values of ( ) b a g , , , ,  the range of  can be easily calculated from (3.9).Similarly for given values of ( ) the range of  can be computed from the following inequality: It is to be mentioned that the conditions under which the suggested class of estimators or equivalently, or equivalently, In similar way for given 0   = , the range of  under which the suggested class of estimator ( ) or equivalently, Further if the values of 0   = is given, then the range of  in which the suggested class of estimators ( )

Comparison with Other Estimators
In SRSWOR scheme, Chami et al ( 2012) proposed a ratio-product-ratio estimator for population mean Y as To the first degree of approximation, the bias and mean squared error of ( ) are respectively given by where It is well known identity that Using (4.4), (4.5), (4.6) and from (2.7) and (4.3), we have ( ) From (2.12) and (4.8) we have Thus the proposed post-stratified estimator is more efficient than the corresponding estimator ( )   , y in SRSWOR scheme at their optimum conditions as long as the condition yx    is satisfied.

The Proposed Separate Ratio-Product-Ratio Estimator Using Auxiliary Information in Post Stratified Sampling
For estimating the population mean Y of the study variable y in post stratified sampling, we define the following separate two-parameter ratio-product-ratio estimator and ( ) Using the results from Stephan (1945) for ( ) to terms of order 1 − n , the bias and MSE of ts y are respectively given by where Inserting  and (0,1) in (5.1) we get the separate ratio and separate product estimators in post stratified sampling respectively as (5.9) (5.10) It is observed from (1.1), (5.4), (5.9) and (5.10) that (i) the separate ratio estimator

( )( )
(iv) the suggested class of separate estimator ts y is more efficient than separate ratio estimator (5.14) (v) the proposed class of separate estimator ts y is more efficient than separate product estimator (5.15) Motivated by Bahl and Tuteja (1991), we consider the separate ratio-type exponential and the separate product-type exponential estimators for population mean Y in post stratified sampling respectively as To the first degree of approximation, the MSEs of ps It is observed form (1.1), (5.9), (5.10), (5.16) and (5.17 Further from (1.17) and (5.4) we have ( ) Similarly for given 0 0 0 0 0 0 and , , , , , the range of Expression (5.34) shows that the difference between

Empirical Study
To exhibit the performance of the proposed estimators, two data sets have been considered.Description of data set is given below:  We also note that from Table 6

1 : 1 :
Population mean of the auxiliary variate x, Population mean of the study variate y, In case of post stratification, usual unbiased estimator of population mean Y is defined as define the coefficient of variation of Y and X as from Stephen (1945), the variance/MSE of ps y to the first degree of approximation is obtained as the sampling fraction.Ige and Tripathi (1989) defined classical ratio and product type estimators for estimating the population mean Y in case of post stratification as , estimate of population means in case of post stratification, mean of the auxiliary variate x and h x is the mean of the sample of size h n that fall in the th h stratum.MSE of the Ige and Tripathi (1989) estimators R ps Y ˆ and P ps Y ˆ are


coefficient of variation of x in stratum h.Motivated byKhoshnevisan et al (2007), we define a class of separate ratio-type estimators for population mean Y in post stratified sampling as or functions of known parameters of the auxiliary variable x in the h th stratum of the population such as standard deviation of the h th stratum.To the first degree of approximation the MSE of ps SR obtained from (1.17) just by putting 4) to zero, we obtain

.
the values of  from (2.5), becomes an (approximately) unbiased estimator for the population mean Y .In the three dimensional parameters space When the sample size n is sufficiently large (i.e.n approaches the population size N) the bias of 10) into the estimator, an asymptotically optimum estimator (AOE) post stratified sample mean ps y is preferred.Therefore combining the conditions (i)-(iv) with the condition the following explicit ranges.
It is well known that the ratio estimator R ps y is more efficient than the usual unbiased estimator It is well known that the product estimator P ps y is more efficient than the usual unbiased estimator 5) and (2.7) we have the Onyeka (2012) class of estimators pss y just by putting the appropriate values of the constants

3. 5
Comparison of the proposed class of estimators range of in which the proposed class of estimators than Tailor et al (2017) ratio-type exponential estimator ps Y Re ˆis: Tailor et al (2017) ratio-type estimator ps Y Re ˆis: 16) Thus from (3.14) and (3.16) we can calculate the range of  for given 0   = and the range of  for given 0   = .

3. 6
Comparison of the proposed class of estimators pre-assigned constant.Then the range of  in which the suggested class of estimators et al (2017) product-type exponential estimator ps Pe Y ˆis given by: Tailor et al (2017) product-type exponential estimator ps Pe Y ˆis given by 22) Thus from (3.21) and (3.22) we can easily get the range of  for given 0   = and the range of  for given 0   = .
7)Expression (4.7) clearly indicates that the post-stratified estimator equals to the approximate MSE of the regression estimator regression coefficient of y on x,

Theorem 5 . 1 -
To the first degree of approximation,

Y
and (0,1) yield the MSEs of to the first degree of approximation respectively as

A
20) and(5.22)we get the separate ratio-type exponential estimator ps s 5.21) and (5.23) it is observed that the separate product-type exponential estimator ps Two Parameter Ratio-Product-Ratio Estimator in Post Stratification 290 from (5.4), (5.18) and(5.19)that the proposed class of separate estimators ts y performs better than the separate ratio-type exponential estimator ps s be given.Then the range of h  under which the proposed class of estimators ts y is more efficient than the class of estimators ps SR Y ˆ:

h
in which the suggested class of estimators ts y is more efficient than the class of estimators ps SR Y ˆif

(
the suggested class of estimators ts y is more efficient than the class of estimators ps SR Y ˆand hence the members (proposed by Bacanli and Aksu (2012)) of the ps Rs Y ˆ-family of estimators.Now from (4.3) and (5.4) we have

.
It follows that the separate class of estimators ts y is more efficient than the combined class of estimators ( ) that unless the regression coefficient is same from stratum to stratum in the separate class of estimators ts y is better than the combined class of estimators at optimum condition.
efficiency for both the data sets.It is further observed that the proposed estimators are more efficient than ps y , exact optimum values.Thus there is enough scope of selecting the values of scalars

1 Comparing the MSE of the Usual Unbiased Estimator ps y to the Suggested Estimator
ps y , ratio estimator R ps y ˆ and product estimator P ps y ˆ.A Two Parameter Ratio-Product-Ratio Estimator in Post Stratification 280 3.

Table 6 .
4 gives the optimum values of h

Table 6 .
5gives the PRE of the proposed separate ratio-product-ratio estimator ts y with respect to usual estimator

Table 6 . 1 :
PRE of different estimators with respect to usual unbiased estimator ps y .

Table 6 . 2 :
Optimum values of  for given  .

Table 6 . 3 :
PRE of the proposed combined ratio-product-ratio estimator with respect to ps y , R

Table 6 . 4 :
Optimum values of h

Table 6 . 5 :
PRE of the proposed separate ratio-product-ratio estimator with respect to

.75,0.75,1.29,1.42)
It is observed from Tables 6.3 and 6.5 that the proposed combined and separate estimators are more efficient than .1 that the proposed optimum separate estimator ) Thus we recommend our proposed estimators for their use in practice.