New Ratio-Type and Product-Type Estimators Using Systematic Sampling Design

Introduction

Use of the auxiliary variables is very important for the construction of the efficient estimators for the estimation of population parameters under different sampling designs. The importance of systematic sampling cannot be overemphasized, being one of the sampling schemes widely used in practice due to its simplicity. Systematic sampling has got the nice feature of selecting the whole sample with just one random start. Apart from its simplicity, which is of considerable importance, this method yields certain estimators which are more efficient than simple random sampling and stratified random sampling for certain types of populations, see Cochran (1946), Gautschi (1957), Hajeck (1959). Consider a finite population 𝑈 = {𝑈 1 , … . , 𝑈 𝑁 } of size 𝑁 units, numbered from 1 to 𝑁. Let us assume that 𝑁 = 𝑛𝑘, where 𝑛 and 𝑘 are positive integers. Thus, there will be 𝑘 samples, each of size 𝑛. Observe the study variate 𝑦 and the auxiliary variates 𝑥 and 𝑧 for each and every unit selected in the sample. Let 𝑦 𝑖𝑗, 𝑥 𝑖𝑗 , 𝑧 𝑖𝑗 ; 𝑖 = 1,2, … . , 𝑘; 𝑗 = 1,2, … . 𝑛 denote the value of 𝑗𝑡ℎ unit in the 𝑖𝑡ℎ systematic sample. Let the systematic sample means be defined as 𝑦 ̅ 𝑠𝑦 = , respectively and 𝐶 𝑦 , 𝐶 𝑥 , 𝐶 𝑧 being the coefficients of variation of the variates 𝑦, 𝑥, 𝑧, respectively.

To obtain the properties of the estimators up to first order of approximation, let us consider

such that 𝐸(𝑒 𝑖 ) = 0, for 𝑖 = 0,1 and 2. Then we have

The variance of the usual unbiased estimator for the population mean in systematic sampling is given by

The ratio and product estimators for finite population mean suggested by Swain (1964) and Shukla (1971) are, respectively, given by

and 𝑦 ̅ 𝑃𝑠𝑦 = 𝑦 ̅ 𝑠𝑦 ( 𝑧̅ 𝑠𝑦 𝑍 ̅ ).

The mean square errors of the estimators 𝑦 ̅ 𝑅𝑠𝑦 and 𝑦 ̅ 𝑃𝑠𝑦 , to the first order of approximation, are, respectively, given as .

New Ratio-Type Estimator Under Systematic Sampling

We suggest the following ratio-type estimator for the population mean using systematic sampling design as

where 𝛼 is a real constant to be determined such that the 𝑀𝑆𝐸 of 𝑦 ̅ 𝑅𝑒𝑠𝑦 ′ is minimized. It is observed that the estimator reduces to 1. usual unbiased estimator 𝒚 ̅ 𝒔𝒚 for 𝛼 = 1, 2. Singh et al. (2011) ratio estimator 𝒚 ̅ 𝑹𝒔𝒚 for 𝛼 = 0.

Taking the expectations in (2.1), the 𝑏𝑖𝑎𝑠 of the estimator is obtained as

The 𝑀𝑆𝐸 of the estimator, upto the first order of approximation, is given by

2.1: Optimal Value of the Weight 𝜶

The optimal value of the weight 𝛼, for which the 𝑀𝑆𝐸 of the estimator is minimized, is obtained by using the following condition

On solving (2.1.1), we have

Using the above optimal value of 𝛼 in (2.3) we obtain the minimum attainable 𝑀𝑆𝐸 of the estimator 𝑦 ̅ 𝑅𝑒𝑠𝑦 ′ as

2.2: Efficiency Comparison

For making efficiency comparisons of the estimator 𝑦 ̅ 𝑅𝑒𝑠𝑦 ′ with the existing estimators, we have from (1.1), (1.4), (1.8), (2.

2.3: Numerical Illustration

To examine the efficiency of the proposed estimator over the existing estimators, we have considered the following numerical illustration:

New Product-Type Estimator in Systematic Sampling

We suggest the following product-type estimator for the population mean using systematic sampling design as

where 𝛽 is a real constant to be determined such that the 𝑀𝑆𝐸 of 𝑦 ̅ 𝑃𝑒𝑠𝑦 ′ is minimized. It is observed that the estimator reduces to 1. usual unbiased estimator 𝒚 ̅ 𝒔𝒚 for 𝛽= 1, 2. Singh et al. (2011) ratio estimator 𝒚 ̅ 𝑷𝒔𝒚 for 𝛽 = 0.

Taking the expectations in (3.1) the 𝑏𝑖𝑎𝑠 of the estimators is obtained, respectively, as

The 𝑀𝑆𝐸 of the estimator, upto the first order of approximation, is given by

3.1: Optimal Value of 𝜷

The optimal value of 𝛽 , for which the 𝑀𝑆𝐸 of the estimator is minimized, is obtained by using the following condition

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