ln-Type Variance Estimators in Simple Random Sampling

Until now, various types of estimators have been used for estimating the population variance in simple random sampling studies, including ratio, product, regression and exponential-type estimators. In this article, we propose a family of ln -type estimators for the first time in the simple random sampling and show that they are more efficient than the other types of estimators under certain conditions obtained theoretically. Numerical illustrations and a simulation study support our findings in theory. In addition, it has been shown how to determine the optimal points in order to reach the minimum MSE values with the properties of the ln -type estimators in the different data sets.


Introduction
Different types of estimators have been proposed in the literature for estimating the population variance in simple random sampling, including ratio-type ( Kadilar 2013) estimators. Apart from these types of estimators, ln type estimators have used by Cekim and Kadilar (2020) in stratified random sampling methods and Hassan et al. (2020) in two phase sampling methods. ,  and Khan (2015) benefited from transformations of an auxiliary variable; Shabbir and Gupta (2014) and Adichwal et al. (2016) used the information of two auxiliary variables; Singh et al. (2016), Singh and Solanki (2014) and Subramani and Kumarapandiyan (2015) improved estimators as a family for the population variance. The type of estimator used is decided by looking at the relationship between the study variable y and the auxiliary variable x . If the relationship is a straight line passing through the neighborhood of the origin, then ratio and product estimators are equal to regression estimators. Exponential estimators are proposed to make ratio and product estimators more effective than regression estimators when this condition is not satisfied. In this article, we achieve better estimators using the ln -function when the condition is not satisfied. We propose an ln -type estimator of the population variance in simple random sampling for the first time in the literature on Sampling Theory. The usual variance estimator is given by where n is the sample size. Since this estimator is unbiased, its variance expression is given by 2 Isaki (1983) defined the classical ratio estimator to estimate the population variance of the study variable as follows:   The estimator in (3) is motivated for different values of  and  by Upadhyaya and Singh (1999) and Kadilar and Cingi (2006), as presented in Table 1, when the classical ratio estimator is shown as a class by The bias and the MSE of the estimators provided in Table 1 The bias and the MSE expressions of the estimator given in (10) respectively.

Proposed Family of ln -Type Estimators and its Properties
We propose the family of ln -type estimators for the population variance as 22 Table 2 for different values of  and  . We consider the following error terms to obtain the equations of the bias and the MSE of the proposed estimators: By expanding the estimator proposed in (13) and using the notation of  's up to the first order approximation, for 1 After subtracting the population variance from both sides of (14), we obtain

Comparisons
In this section, we compare the MSEs of the proposed estimators

Numerical Illustration
We use the percent relative efficiency (PRE) to evaluate the performance of the considered estimators based on the classical ratio estimator:  For the efficiency comparisons of the mentioned estimators, we apply the data set of Kadilar and Cingi (2006), whose population parameters are given in Table 3. The study variable ( ) Y is the amount of apple production and the auxiliary variable ( ) X is the number of apple trees in all populations. Population 1 is the Marmara region, Population 2 is the Black Sea region and Population 3 is the Edirne city in Turkey. The results in Tables 4 support the theoretical finding that the proposed estimator is the most efficient; the PRE values of the proposed estimator are higher than those of the other estimators for all three populations. Therefore, we recommend using estimators based on the ln -function rather than the exponential function for estimating the population variance. From this result, an optimal point might be found to obtain the minimum MSE for the data sets.

Simulation Study
We perform a simulation study using Population 3 to support the theoretical results and the numerical illustrations. Moreover, the simulation provides some properties of the proposed ln-type estimators. The simulation steps are as follows: (i) Select a sample with size n from the population.  The PRE values of the mentioned estimators over the classical ratio estimator obtained in the simulation study are provided in Table 5. It can be seen that similar to the results of the numerical illustration for Population 3, the proposed estimators have higher PRE values compared to the other estimators, showing that the proposed estimators are more efficient than the others.

Conclusions
The aim of the sampling studies in the literature is to obtain the minimum MSE using various types of estimators. In this article, a family of ln -type estimators was proposed as an alternative to the ratio, regression and exponential estimators. It is found that the proposed estimators achieve a better performance when estimating the population variance in simple random sampling compared to the ratio, regression and exponential estimators in both theory and practice. From this result, we can conclude that using the ln -function in variance estimators improves the efficiency of the estimators. In the future, the proposed estimator can be extended to a family of estimators, such as in the studies of  and Singh et al. (2017). Also, different estimators of other sampling methods can be developed using the ln -function.