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 Sarndal¹ 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 Sarndal⁴, 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 Singh¹⁰, Tracy et al.¹¹, Kim et al.⁶, Clement and Enang², Koyuncu and Kadilar⁷. 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 T of N elements, T = { T₁,T₂,T₃,…,T_N} consists of L strata with N_h units in the h th stratum from which a simple random of size n_h 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 Cochran³, 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:



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.



subject to the constraint



The calibrated weights and the estimator are obtained as show in (2.6) and (2.7) respectively.



As a result, the modified calibrated estimator of Singh et al.⁹ is:



The calibrated estimator's estimated variance is given by



where



is the h^th  strata sample mean square and 



with



Singh¹⁰ introduced new calibration equations to a population mean calibration estimator in stratified sampling. Under stratified sampling, the Singh³ calibration estimator of the population mean Y is given by  



where Ω_h^s is calibrated weight which is chosen so that the Chi-Square distance mean sum is:



is subject to the following constraints as a minimum



The calibrated weight and the estimator (2.11) are obtained as show in (2.15) and (2.16) respectively




The estimated variance of Singh¹⁰ calibrated estimator is given by



where



is the mean square of the hth stratum sample while



Alam et al.¹ developed a calibration estimator for estimating population mean under stratified random sampling using a distance function as follows:



where Ω_h^A are the calibration weights which are chosen with the distance function



where Ω_h^l is the h^{l th} stratum weight which minimum subject to the calibration constraints



Minimization of (2.19) subject to the calibration constraint given in (2.20), the calibration weights are given by



And the calibrated estimator is



The estimated variance of Alam et al.¹ calibration estimator is given by



Proposed Estimator

In stratified random sampling, the conventional Combined Ratio Estimator is given in (2.3) can be written as



Motivated by Alam et al.¹, calibrated combined ratio estimator denoted by



is modified as



where Ω_h denotes the new calibrated weights that minimize the Chi-square distance.



Subject to calibration constraint given by



The Lagrange multipliers technique is employed to compute new calibrated weights (Ω_h), and the following results are obtained:



Differentiating (2.29) partially in relation to Ω_h, and λ, equal to zero



Substitute (2.30) in (2.31), the results are obtained as:



Make λ the subject of formula, obtained as:



On substituting (2.33) in (2.30) the calibrated weights can be written as:



Substituting (2.34) in (2.26), obtain the new combined calibration estimator


as:


Substituting



gives



Setting Q_h = 1 and Q_h = x_h^-1 we have the following new set of calibration combined ratio estimators respectively:



The suggested estimator's estimated variance



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)


 

Table 1: Populations Involved in the Empirical Research Click here to view Table

Table 2: PREs of Some Existing and Suggested Estimators Using Population I. Click here to view Table

Table 3: PREs of Some Existing and Suggested Estimators Using Population II Click here to view Table

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.

Acknowledgements

The authors acknowledge the Associate Editor and anonymous reviewers for their valuable work and recommendations.

Conflict of Interest

The authors declare no conflict of interest.

Funding Sources

The authors received no financial support for the research, authorship and publication of this article.

References

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