Enhanced estimation of population mean using post stratification sampling

Introduction

Stratification is one of the most commonly used techniques in survey sampling. The application of stratified random sampling assumes that the sizes of strata and the sampling frame for each stratum are known beforehand. However, in many practical scenarios, while the overall population size and the proportion of units within different strata are available, the sampling frame for each stratum may be unavailable or difficult to compile due to cost and time constraints. In such cases, stratified random sampling cannot be directly implemented. To address this challenge, the post-stratification technique is used. In this approach, a sample of size n is first drawn using simple random sampling without replacement. This sample is then categorized into different strata and analyzed as a stratified sample. Several researchers have significantly contributed to the development of post-stratification methods, including Holt and Smith (1979) [3] , Jagers et al. (1985) [6] , Jagers (1986) [5] , Ige and Tripathi (1989) [4] , Agrawal and Panda (1993) [1] , and Singh and Ruiz Espezo (2003) [9] , Tailor et al. (2017) [10] . In this paper, we use the data from the auxiliary variable, based on post-stratification sampling, to propose an enhanced estimator for the population mean estimation of the study variable.

This paper objective is to:

 An enhanced estimator for the population mean under post-stratified sampling is proposed to compete with existing estimators.  Comparing the proposed estimators' theoretical efficiency to that of the existing estimators  Use real-world population to illustrate the proposed estimators.

The parts that follow provide further design for the current paper. The methodology and notations utilized are explained in Section 2. Section 3 discusses some existing estimators. In Section 4, we have proposed an enhanced estimator for population mean under post-stratified sampling, along with the bias, MSE and the minimum MSE expressions. The efficiency conditions are obtained in Section 5. Section 6 presents an empirical illustration of the efficiency of the proposed estimator using examples based on real populations. Finally, Section 7 concludes the paper.

Methodology and notations

Let U = ( U 1, U 2 , …….U N ) be a finite population of size N. Using simple random sampling without replacement, a sample of size n is drawn from population U. After selecting the sample, it is observed that which units belong to h th stratum. Let n h be the size of the sample falling in h th stratum such that ∑ n L h=1 h = n. Here it is assumed that n is so large that probability of n h being zero is very small. Let x hi be the observation on i th unit that fall in h th stratum for auxiliary variate x and y hi be the observation on i th unit that fall in h th stratum for study variate y, then

: h th stratum mean for auxiliary variate x,

: h th stratum mean for study variate y

Existing estimators

In case of post-stratification, usual unbiased estimator of population mean 𝐘 ̅ is defined as

is the weight of the h th stratum and

is sample mean of n h sample units that fall in the h th stratum.

Using the results from Stephen (1945), the variance of 𝐲 ̅ 𝐏𝐒 to the first degree of approximation is obtained as

Ige and Tripathi (1989) [4] defined classical ratio type and product type estimators for estimating the population mean Y ̅ in case of post stratification as

is the unbiased estimate of population mean in case of post stratification,

is the known population mean of the auxiliary variate x and x ̅ h is the mean of the sample of size n h that fall in the h th stratum.

To the first degree of approximation MSE of estimators 𝐘 ̅ ̂𝐏𝐒

𝐑 and 𝐘 ̅ ̂𝐏𝐒 𝐏 are respectively given by

Tailor et al (2017) [10] suggested the following combined ratio-type and product type exponential estimators for population mean of Y ̅ in post-stratified sampling as

To the first degree of approximation, the MSEs of 𝐘 ̅ ̂𝐏𝐒 𝐑𝐄 and 𝐘 ̅ ̂𝐏𝐒 𝐏𝐄 are respectively given by

Proposed Estimator

Taking clue from Kumar and Siddiqui (2024) we suggest a class of estimator for population mean Y ̅ of y under post stratified sampling as

Where k j, j = 1,2 are suitable chosen constants and η and λ are either real amount or parametric values associated with auxiliary variable/ study variable To obtain the bias and MSE of the proposed class of estimator 𝐲 ̅ 𝐏𝐒 * we write

And to the first degree of approximation

Now expressing 𝐲 ̅ 𝐏𝐒 * in terms of 𝐞 𝟎 𝐚𝐧𝐝 𝐞 𝟏 we have

Expanding the right hand side of (4.2), multiplying out and neglecting terms of 𝐞 𝐢 ′ 𝐬 having power greater than two we have

Taking expectation of both sides of (4.3) we get the bias of 𝐭 as

Squaring both sides of (4.3) and neglecting terms of 𝐞 𝐢 ′ 𝐬 having power greater than two we have

Minimizing the 𝐌𝐒𝐄( 𝐲 ̅ 𝐏𝐒 * ) against 𝐤 𝟏 provides the optimum value of k1 as

Using the value of 𝐤 𝟏(𝐨𝐩𝐭) in the 𝐌𝐒𝐄( 𝐲 ̅ 𝐏𝐒 * ) we get the minimum MSE of 𝐲 ̅ 𝐏𝐒 * as

Putting 𝐤 𝟏 = 1 in (4.6) we have

Minimizing the MSE of 𝐲 ̅ 𝐏𝐒 * against 𝐤 𝟐 we get the optimum value of 𝐤 𝟐 as

Substitution of (4.10) in (4.9) yield the minimum MSE of 𝐲 ̅ 𝐏𝐒 * (at k1 = 1) as

Which equals to the minimum MSE of the difference estimator y ̅ d = y ̅ PS + d(X ̅ -x ̅ PS ), d being a constant

Efficiency conditions

By comparing the MSE( y ̅ PS * ) min to the variance/MSE/minimum MSE of the estimators we get the conditions under which suggested estimator y ̅ PS * is better than existing estimators.

MSE( y ̅ PS

From (5.1) to (5.5) it follows that the proposed estimator y ̅ PS * is more efficient than the estimators Y ̅ ̂PS R , Y ̅ ̂PS P , Y ̅ ̂PS RE and Y ̅ ̂PS PE as long as the conditions (5.1), (5.2), (5.3), (5.4) and (5.5) are satisfied respectively.

Numerical study

To strengthen the theoretical establishment, we have conducted an empirical investigation on a real population. The descriptive statistics of this population are presented in Table 1. The optimum values of η and λ are computed and the ranges of η (for fixed values of λ) and λ (for fixed values of η) are also computed and the results are presented in Tables 2, 3 and 4, respectively. In these tables, for different fixed values of η (such as = 0.25, 0.5, 1) and varying values of λ (such as λ=-1,-2,-3,-4,-5) are considered to evaluate the performance of the proposed estimator y ̅ PS * . It is observed that under these combinations, the proposed estimator y ̅ PS * performs more efficiently than the existing estimators.

Population: Onyeka (2012, 65-78) [11]

Conclusions

This paper proposes an enhanced estimator for the population mean using post stratified sampling. The bias, mean square error, and minimum mean square error of the proposed estimator are obtained mathematically to the approximation of first order. By comparing the minimum MSE of the proposed estimator with the MSE of the existing estimators, the efficiency conditions are determined. A numerical illustration using real population data along with empirical results, demonstrate that the proposed estimator outperforms conventional estimators.

References

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