A Generalized Class of Exponential Type Estimators for Population Mean under Systematic Sampling Using Two Auxiliary Variables

We have proposed a generalized class of exponential type estimators for population mean under the framework of systematic sampling using the knowledge of two auxiliary variables. The expressions for the mean square error of the proposed class of estimators have been corrected up to first order of approximation. Comparisons of the efficiency of the proposed class of estimators under the optimal conditions with the other existing estimators have been presented through a real secondary data. The statistical study provides strong evidence that the proposed class of estimators in survey estimation procedure results in substantial efficiency improvements over the other existing estimation approaches.


Introduction
In the literature of survey sampling, it is well known that the efficiencies of the estimators of the population parameters of the variable of interest can be increased by the use of auxiliary information related to auxiliary variable , which is highly correlated with the variable of interest .Auxiliary information may be efficiently utilized either at planning stage or at design stage to arrive at an improved estimator compared to those estimators, not utilizing auxiliary information.A simple technique of utilizing the known knowledge of the population parameters of the auxiliary variables is through ratio, product, and regression method of estimations using different probability sampling designs such as simple random sampling, stratified random sampling, cluster sampling, systematic sampling, and double sampling.
Let us consider a finite population  of size  of distinct and identifiable units,  1 ,  2 ,  3 , . . .,   and number it from 1 to  unitsin some order.A random sample of size  units is selected from the first  units and then every th subsequent unit is selected; thus there will be  samples (clusters), each of size  and observe the study variable  and auxiliary variable  for each and every unit selected in the sample.Let ( ,   ) for  = 1, 2, . . .,  and  = 1, 2, . . ., : denote the value of th unit in the th sample. where are the corresponding intraclass correlation coefficients for the study variable  and the auxiliary variables  and , respectively.
Similarly   =   /    ,   =   /    , and   =   /    are the correlation coefficients of the study and the auxiliary variables, respectively, where   ,   , and   are the population standard deviation of study variable  and auxiliary variables  and , respectively.Also   ,   , and   are the population covariances between  and ,  and , and  and , respectively.Also let   and   and   be the population coefficients of variation of the study and the auxiliary variables, respectively.
The variance of the classical estimator unbiased estimator  1 is given by where  = (( − 1)/).Swain [7] proposed a ratio estimator in systematic sampling given by The mean squared error of the above estimator is as follows: where  * * =  *  / *  and  =     /  .Shukla [9] suggested the following product estimator for population mean of the study variable;the suggested estimator and their mean squared error are given as follows: where  * * 2 =  *  / *  and  * =     /  .
The usual regression estimator for population mean under systematic sampling is given as follows: where   is the sample regression coefficient between  and .
The variance of the estimator  4 , up to first order of approximation, is as follows: Singh et al. [17] recommended ratio-product type exponential estimators and are given by The mean square errors of the Singh et al. [17], using first order of approximation, are given as follows: Tailor et al. [20] suggested a ratio-cum-product estimator for finite population mean; the recommended estimator and their first order mean square error are shown as follows: where

The Generalized Class of Exponential Estimators
In this section, we have proposed a generalized class of exponential type estimators for population mean of the study variable , under the framework of systematic sampling as given by where −∞ <  < ∞, −∞ < ℎ < ∞,  > 0, and  > 0.
A set of some new and known members of the generalized class of exponential estimators generated from (14) for some suitable values of , ℎ, , and  are listed in Table 1.
To obtain the properties of the proposed class of estimators up to first-order approximation, we use the following relative errors, symbols, and notations: such that also Expanding ( 14) in terms of 's up to the first order of approximation, we have ) . ( Further simplify where and  3 = ℎ/.On squaring and taking expectation on both sides of ( 19), we get the mean square error of   , up to the first degree of approximation, as where  1 = / and  2 = ℎ/.By partially differentiating (20) with respect to  1 and  2 , we get the optimum value of  1 and  2 as given by where Substituting the optimal values of  1 and  2 in (20) we obtain the minimum mean square error of the estimator   as follows:

Comparison of Efficiency
In this section, we have found some theoretical efficiencies conditions under which the proposed estimator performs better than the other relevant existing estimators by comparing the generalized class of exponential type estimators with other existing estimators.

Empirical Study
To examine the merits of the proposed estimator over the other existing estimators at optimum conditions, we have considered natural population data sets from the literature.The sources of population are given as follows.

Conclusion
In this paper we proposed a generalized class of exponential type estimators for the population mean of study variable , when information is available on two auxiliary variables under the framework of systematic sampling scheme.The properties of the proposed estimator are derived up to first order of approximation.The proposed estimator is compared with other present estimators, both as theoretical and empirical efficiency comparisons.We have also judged the performance of the proposed estimator for a known natural population dataset; see Tailor et al. [20].Results are given in Table 2 which shows that performances of the proposed generalized class of exponential type estimator are more efficient than the other existing estimators by smaller mean square errors and the higher percent relative efficiencies of the estimators.Hence it is preferable to use the proposed estimator in practical surveys.

Table 1 :
Some members of the proposed class of estimators.

Table 2 :
The mean square errors (MSEs) of the estimators and the percent relative efficiencies (PREs) with respect to  1 .