Calibration of Stratified Random Sampling with Combined Ratio Estimators Oriental Journal of Physical Sciences

This study considered modification of combined ratio type calibration estimators in stratified random sampling using calibration estimation approaches. The estimators of population mean in stratified random sampling depends on the strata estimated sample means. However, the means are sensitive to the extreme values or outliers in the sample observations of the study variables and strata sizes respectively. A new sets of calibration weights and property of the suggested combined calibration estimators of population mean in stratified sampling were derived. Empirical study through simulation was conducted to investigate the efficiency of the modified combined ratio-type calibration estimators of population mean obtained and the results revealed that the suggested estimators of population mean performed better than some existing estimators considered in the study.


Introduction
Proper utilization of auxiliary information to obtain the efficiency of estimates of the population mean has increased in the theory of sample surveys. Many researchers have used auxiliary information in product, ratio and regression type estimators to obtain more efficient estimator under different sampling scheme. Calibration resolution is used in stratified random sampling in order to achieve optimum strata weights for precision improvement of estimates of parameters. In calibration estimation, new stratum weights are calculated to minimize a certain distance measure from the original design weights while meeting a set of auxiliary information restrictions. Deville and Sarndal 1 established the approach of estimate by calibration in survey sampling in 1992. The concept is to employ auxiliary data (auxiliary information) to improve a population statistic estimate. Following Deville and Sarndal, 4 Singh et al. 9 was the first to extend a method of calibrating to a stratified sampling design. Many other researchers have investigated calibration estimates in survey sampling design utilizing various calibration constraints. These researchers include Singh, 10 Tracy et al., 11 Kim et al., 6 Clement and Enang, 2 Koyuncu and Kadilar. 7 In stratified sampling, Rao et al. 8 suggested a multivariate calibration estimator for the population mean based on different distance measures and two auxiliary variables. In the previous studied, none have considered calibration estimation in combined ratio estimators. In this study, calibration approaches have been adopted in combined ratio estimator with aim to obtain highly efficient estimators of population mean in stratified random sampling. The presence of extreme values in the observation of the study variable have no or little effect on the other estimates.
Take a look at a finite population of elements, T={T 1 , T 2 , T 3 ,...,T N consists of L strata with N h units in the hth stratum from which a simple random of size can be generated from the population using SRSWOR. Total Population size , sample size where y hi , i=1,2..., N hi and x hi , i=1,2,..., N hi of auxiliary variable x and study variable y. Let W h = N h / N be the weights of the strata, the sample mean and population mean for the study variable.

Literature Review
According to Cochran, 3 with stratified sampling, the classic estimator of population mean is given as: Hansen et al. 5 suggested a combined ratio estimator as The combined ratio estimator's variance is given as follows: ... (2.4) where Singh et al. 9 presented the calibration approach for the combined general regression estimator (GREG) for the population mean given by By minimizing the Chi-Square distance measure, Singh et al. 9 were able to get new calibration weights.
... (2.6) subject to the constraint The calibrated weights and the estimator are obtained as show in (2.6)  where is the hth strata sample mean square and .

Empirical Study Using Simulation
In this section, a simulation research was carried out to see if the proposed estimators were better than the other estimators evaluated in the study.
For this investigation, 1000-unit data was generated.
Using the function defined in Table 1, populations were stratified into three non-overlapping heterogeneous groups of 200, 300, and 500. Method SRSWOR was used to randomly choose samples of sizes 20, 30, and 50 from each stratum 10,000 times. The precision (PRE) of the estimators under consideration was calculated using (2.40) ...(2.40)    Tables 2 and 3 shown the PREs of some existing and suggested estimators considered in this study using data generated by populations I and II respectively. From the results obtained, it revealed that the suggested estimators outperformed the existing estimators considered in the study.

Conclusion
From the results obtained, the empirical study revealed that on the efficiency of the suggested calibration estimators versus the study's current related estimators, the suggested estimators have higher PREs compared to some existing calibration estimators in the numerical analysis. The suggested estimators outperformed other calibration estimators because the suggested estimators demonstrated high level of efficiency over other estimators. Hence, the suggested estimators are closers to the true values of the population mean compared to other existing calibration estimators in which the suggested estimators have more chances of producing estimates that are closer to the population mean's true value.