Journal of Probability and Statistics

Volume 2016 (2016), Article ID 2374837, 6 pages

http://dx.doi.org/10.1155/2016/2374837

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

Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan

Received 26 June 2016; Accepted 31 July 2016

Academic Editor: Z. D. Bai

Copyright © 2016 Mursala Khan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. 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.

In the present paper we will use knowledge of the auxiliary variables under the framework of systematic sampling. Due to its simplicity, systematic sampling is the most commonly used probability design in survey of finite populations; see W. G. Madow and L. H. Madow [1]. Apart from its simplicity, systematic sampling provides estimators which are more efficient than simple random sampling or stratified random sampling for certain types of population; see Cochran [2], Gautschi [3], and Hajeck [4]. Later on the problem of estimating the population mean using information on auxiliary variable has also been discussed by various authors including Quenouille [5], Hansen et al. [6], Swain [7], Singh [8], Shukla [9], Srivastava and Jhajj [10], Kushwaha and Singh [11], Bahl and Tuteja [12], Banarasi et al. [13], H. P. Singh and R. Singh [14], Kadilar and Cingi [15], Koyuncu and Kadilar [16], Singh et al. [17], Singh and Solanki [18], Singh and Jatwa [19], Tailor et al. [20], Khan and Singh [21], and Khan and Abdullah [22].

Let us consider a finite population of size of distinct and identifiable units, 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 and : denote the value of th unit in the th sample. Then the systematic sample means are defined as and are the unbiased estimators of the population means and , respectively.

Further letwhereare 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 is given bywhere .

Swain [7] proposed a ratio estimator in systematic sampling given byThe 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 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 , up to first order of approximation, is as follows:Singh et al. [17] recommended ratio-product type exponential estimators and are given byThe 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 .

#### 2. 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 bywhere , , , and .

A set of some new and known members of the generalized class of exponential estimators generated from (14) for some suitable values of , , , and are listed in Table 1.