A Two-Parameter Ratio-Product-Ratio Type Exponential Estimator for Finite Population Mean in Sample Surveys

This paper suggests a two-parameter ratio-product-ratio type exponential estimator for a finite population mean in simple random sampling without replacement (SRSWOR) following the methodology in the studies of Singh and Espejo (2003) and Chami et al (2012). The bias and mean squared error of the suggested estimator are obtained to the first degree of approximation. The conditions are obtained in which suggested estimator is more efficient than the sample mean, classical ratio and product estimators, ratiotype and product type exponential estimators. An empirical study is given in support of the present study.


Introduction
In sample surveys it is well established fact that the improvement in the precision of an estimator of the population mean Y of the study variable y is possible by using information on an auxiliary variable x (highly correlated with the study variable y ) at the estimation stage.It is known that if the correlation between study variate y and the auxiliary variate x is positive (high) the usual ratio estimator is employed.On the other hand if this correction is negative (high), the product method of estimation can be employed.Consider the finite population (1) The classical ratio and product estimators for population mean Y are respectively given by and x y y P (1.3)It can be shown that the ratio estimator R y is more efficient than the unbiased estimator y if (1.7) The ratio-type exponential estimator Re y is better than unbiased estimator y if obtained in which the proposed class of ratio-product-ratio-type exponential estimators is preferred.We carry out an empirical study showing that the proposed class of ratioproduct-ratio-type exponential estimators out performs the estimators y ,

The Suggested Two Parameter Ratio-Product-Ratio Type Exponential Estimator
For estimating the population mean Y of the main variable y , we suggest the following two-parameter ratio-product-ratio type exponential estimator: where  , are real constants.The aim of this paper is to derive values for these constants  , such that the bias and / or the mean squared error (MSE) of ( )  , e T is minimal.
We mention that ( ) , that is the estimator is invariant under point reflection through the point ( ) . In the point of symmetry ( ) , the estimator reduces to the sample mean y , that is, we have

Bias and Mean Squared Error (MSE) of the Proposed Estimator
Applying the standard techniques we evaluate the first degree of approximation (upto terms of order 1 − n ) to the bias and mean squared error (MSE) of the suggested estimator ( ) is the sampling fraction.Further, it is assumed that the sample is so large as to make 0 e and 1 e small, justifying the first degree approximation considerd wherein we ignore the terms involving 0 e and/or 1 e in a degree greater than two, see Sahai and  Ray(1980, p.272).

Bias of the Estimator
Expanding the right hand side of (2.2) and neglecting terms of e's having power greater than two we have Taking expectation of both sides of (2.4) we get the bias of ( ) to the first degree of approximation as The bias of ( ) The suggested ratio-product -ratio-type exponential estimator ( ) , substituted with the values of  from (2.6), becomes an (approximately) unbiased estimator for the population mean Y ˆ.In the three dimensional parameter space ( , these unbiased estimators lie on a plane (in the case ).We mention further that as the sample size n approaches the population size N, the bias ( ) becomes negligible, since the factor ( ) Mean Squared Error of ( )   , e T Squaring both sides of (2.4) and neglecting terms of s ' e having power greater than two we have Taking expectation of both sides of (2.7) we get the MSE of the estimator ( ) to the first degree of approximation as (2.9) Equating (2.9) to zero to obtain the critical points, we get the following solutions: , in which case we get a local minimum.However, the critical points determined by (2.11) are always local minima; for a given C , (2.11) is the equation of a hyperbola symmetric through ( ) gives the unbiased estimator y (sample mean) of the population mean Y .Thus we get the MSE of the sample mean y as (2.12) For  ( ) whose bias and MSE to the first degree of approximation are respectively given by In fact Srivastava (1971Srivastava ( , 1980) ) has shown that ( ) ( ) is the minimal possible MSE up to first degree of approximation for a large class of estimators to which the estimator (2.1) also belongs, for example, for estimators of the form . Further Srivastava and Jhajj (1981) incorporating sample and population variances of the auxiliary variable x might yield an estimator that has a lower MSE than ( ) ( )   ( )

Comparison of Mean
Combining these with the condition we get the following explicit ranges : We note that the case C=0 implies , 0 =  and thus the sample mean y is the estimator with minimal MSE.Putting ( ) ( )

8) we get the MSE of the ratio-type exponential estimator
Re y as From (2.8) and (3.2) we have , we have the following: we obtain the following:

Comparing the MSE of the Product-Type Exponential Estimator
We obtain the following two cases: We mention that this implies and the range for  and  where these inequalities hold are explicitly given by the following two cases: For any given C, we again note that the two regions determined here are symmetric through ( )


. We also note that the parameters ( )   , which yields an AOE [see (2.11)] which for a fixed C lie on a hyperbola, are contained in these regions.
In case (ii), where (and therefore automatically ), the range of  and  are given by

Comparing the MSE of the Classical Ratio Estimator R y to the Suggested Estimator
To the first degree of approximation, the MSE of the usual ratio estimator R y is given by From (2.8) and (3.9) we have Hence from solution (i), where , C 1  we have the following: we obtain the following: Comparing the MSE of the classical Product Estimator P y to the Suggested Estimator From (2.8) and (3.12) we have Thus the proposed estimator is more efficient than the classical product estimator (3.15) or equivalently,

Unbiased Asymptotically Optimum Estimator (AOE)
From (2.6) and (2.11), we can calculate the parameters  and  where our proposed estimator becomes at least up to first approximation-an unbiased AOE.We obtain a line with 2 1 =


(recall that on this line our estimator always reduces to the sample mean estimator y ) in the parameter space with ( ) We mention that the parametric "curve" in (4.2) is only defined for 0 fact, this parametric "curve" is three hyperbolas.
Inserting the values of ( ) ( ) 2) in (2.1), the proposed estimator takes the form: It can be easily shown to the first degree of approximation that .Reddy (1978) has shown that the value of 'C' is more stable than other population parameters such as, the linear regression coefficient, over a period of time and least affected by the sampling fluctuations.So its value can be quite accurately guessed from the past data or a pilot survey or experienced gathered in due course of time.However, if the value of 'C' is not known, one may estimate it on the basis of sample observations without much loss of efficiency, for instance, see Singh et al (1994, p. 216).

Empirical Study
To illustrate the performance of the proposed class of estimators with respect to y , Re y , Pe y , R y and P y , using the following formulae: estimate the population means of the variates ( ) x , y respectively.It is desired to estimate the population mean Y of the study variable y using known population mean X of the auxiliary variable x.For estimating the population mean Y , a simple random sample of size n is drawn without replacement from the population U.The usual unbiased estimator for population mean Y is defined by (1991)  suggested the ratio-type and product-type exponential estimators for the population Y respectively as

9 )
In this paper taking motivation fromSingh and Ruiz Espejo (2003) andChami et al (2012), we have suggested a class of ratio-product-ratio-type exponential estimators for estimating the population mean Y and its properties are studied under large sample approximation.We have compared the proposed class of ratio-product-ratio-type exponential estimators with the three traditional estimators ( ) this context reader is referred to Shirley et al (2014), Singh and Pal (2015, 2017) Singh et al (2016) and Sharma and Singh (2015).
Bahl and Tuteja (1991) product-type exponential estimator Pe y defined at (1.7).Owing to the simplicity of the proposed estimator ( ) obtained from it by choosing appropriate parameters why we study the estimator in (2.1) and compare it to the three known estimators ( y , in terms of e's we have

(
estimator for the population mean Y as ( ) relationship between the study variate y and the auxiliary variate x is markedly non-linear.Thus irrespective of value of C, we are always able to select an AOE

Squared Errors and Choice of Parameters 3 . 1
Comparing the MSE of the Sample Mean y to the Suggested Estimator ( )

1 
the ratio-type exponential estimator Re y due toBahl and Tuteja (1991) is used instead of the sample mean y or product-type exponential estimator Pe y .Thus, we are concerned with a range of plausible values for  and  , where the suggested estimator (2.9)  we get the MSE of the product-type exponential estimator Pe y to the first degree of approximation as , seek a range of  and  values, where the suggested estimator ( )

(
, we are concerned with a range of plausible values for  and  where the proposed estimator ( )The MSE of the usual product estimator p y to the first degree of approximation as

 5 ) 2 1Remark 5 . 1 :
of (4.3) is an unbiased AOE.Using(2.11)in(2.5)we get the first degree of approximation of the bias of an AOE It follows from (4.5) and (2.11) that the bias can only be made zero if 0 least possible bias.Given(2.11)  and unless C=0, we can only assume  be close to and select  accordingly.It is important to mention that the optimum choice of


estimator y , Bahl and Tuteja's (1991) ratio-type ( ) Re y and product-type ( ) Pe y exponential estimators, classical ratio estimator R y and product estimator P y we have two natural population data sets.Population-I [Source: Kadilar and Cingi (2003)] y: Apple production amount.x: Number of apples trees.Population of irrigated area x: Area under crop gram and mixture during 1983-1984 in a village of Rajasthan 400 = We have computed the percent relative efficiency (PRE) of the proposed class of estimators (

5 ) 1 :Note 2 :
NoteWe have computed the values of population I as the correlation coefficient between study variate y and auxiliary variate x is positive.We have computed the values of as the correlation between the study variable y and variable x is negative.Findings are shown in tables 6.1 and 6.2.