#### Abstract

Population mean is an important characteristic from many perspectives in statistical methodology. In this connection, many separate ratio type estimators of population mean have been developed by the researchers to gain more precision and efficiency available in the literature. This study suggests a new class of logarithmic separate ratio-product type estimators of a finite population mean in stratified random sampling using one auxiliary variable to achieve enhanced precision and efficiency. The mean square error expression for the proposed class of estimators and some selected estimates are derived till first order of approximation. Three real data sets and a simulation study reveal that the proposed estimators outperform other estimators used in this analysis.

#### 1. Introduction

Researchers developed many estimators for mean in different scenarios. This study focuses on mean estimation in stratified random sampling. In the literature, there are many ratios, products, regressions, and exponential estimators for the population mean in stratified random sampling. Haq and Shabbir [1] introduced two improved family of estimators for population mean in simple as well as in stratified random sampling. For population mean in stratified random sampling, Kadilar and Cingi [2, 3] suggested some ratio estimators. Koyuncu and Kadilar [4, 5] proposed mean estimators using single as well as two auxiliary variables in stratified random sampling. In the presense of nonresponse, Kumar and Bhougal [6] proposed estimator of the population mean. In stratified random sampling using two auxiliary variables, Malik and Singh [7] proposed the estimator of population mean. Using information on sensitive variable in stratified double phase sampling, Mushtaq et al. [8] suggested estimator of the population mean. Pandey [9] introduced product-cum-power estimators of the population mean. In the presence of nonresponse under stratified two-phase random sampling, Sanaullah et al. [10] proposed a generalized exponential type ratio-cum-ratio and product-cum-product estimators for population mean. In simple as well as in stratified random sampling, Shabbir and Gupta [11, 12] proposed some new estimators of population mean, and Singh et al. [13] proposed exponential type estimator of population mean [14].

Till date many ratios, products, exponential, and regression type estimators have been developed. Developing logarithmic type estimators is relatively new in the field. Izunobi and Onyeka [15] utilized the concept of logarithm and developed logarithmic type ratio and product estimators of the population mean in simple random sampling technique. They compared these logarithmic type estimates with the customary ratio and exponential type estimates. Their newly developed estimators turned out to be the most efficient among all three. A separate ln-type type estimator of variance in stratified random sampling was introduced by Cekim and Kadilar [16] as an alternate to another type of estimators used in their study. They concluded that their proposed estimator was the best alternative. In the same fashion, Cekim and Kadilar [17] introduced a family of logarithmic type estimators of variance in simple random sampling. Their developed estimators turned out to be the efficient ones as compared to the ratio, regression, and exponential type estimators. They also concluded that the use of the “ln function” in the proposed estimators, improved the efficiency of the estimators.

However, less work has been done in the literature deriving logarithmic type estimators for the mean. The aim of this paper is to develop more precise and efficient logarithmic ratio-product type estimator for the first time for the population mean in stratified random sampling.

In order to describe our proposed estimator, we proceed as follows:

Let the finite population ψ = {ψ1, ψ2, …, ψN} consisting of units which are divided in to “d” strata. Let the dth (d = 1, 2, 3, …, Z) stratum contains units such that A sample of size from each stratum is drawn without replacement as well as independently under simple random sampling such that  = ц. Let the sample mean estimates of response and auxiliary variables be  =  and  =  respectively. The corresponding population characteristics will be  =  and  = , respectively. The variances of response and auxiliary variables are  =  and  =  and is the covariance between X and Y, respectively. Furthermore, we assume  = ,  = ,  =  where E () = E() = 0, E() = , E () = , E () = . represents the correlation between the values of X and Y variables of the th stratum, represents the coefficient of variation of the auxiliary variable in the dth stratum. is the coefficient of kurtosis of x variable in the dth stratum.

#### 2. Some Selected Estimators of Variance from the Literature

(i)The separate ratio estimator is as follows:Expressing in terms of ’sIgnoring higher order terms of ’s, we getUpon squaring the above equation, we get MSE value as below:where  = цd/ and  = .(ii)A stratified version of Bahl and Tuteja [18] exponential is as follows:Expressing in terms of ’sSquaring both sides and then applying expectation we get(iii)Motivated by Sisodia and Dwivedi [19] and Kadilar and Cingi [3], Tailor and Lone [20] proposed the following estimators:Expressing it in terms of ’sSquaring both sides and applying expectation, we get MSEIn the same fashion, the MSE expressions for the rest of the three estimators are obtained.

#### 3. The Proposed Estimator

Inspired by Iznobi and Onyeka [15] and Kumar and Boughal [6], we suggest a general class of logarithmic ratio-product type estimator:

where, a and b takes the values , and respectively. Furthermore, when  = 1, reduces to a customary ratio type estimator, i.e.,  =  .

Also, when  = 0, reduces to a stratified version of product type estimator by Iznobi and Onyeka [15], i.e., .

The class of proposed estimators is as follows:

In order to find the MSE expressions, we write in terms of i’s as follows:

Squaring both sides and applying expectation also ignoring higher powers of we get MSE expression as follows:

For finding the optimum value of MSE, we differentiate MSE() w.r.t. and equating it to zero, i.e., MSE() = 0.

On simplification, we get

Putting this value of λ in the expression for MSE(), we get

This is the optimum value of MSE() for i = 1,2,3,4,5.

#### 4. Numerical Results

To evaluate the performance of the suggested proposed estimator, we consider three data sets.

Dataset 1. Source: Singh and Manghat [21], page 208.
Y: expected sale for the current summer, X: the number of refrigerators sold during last summer.

Dataset 2. Source: Murthy [22], Page 228.
Y: output for the factories in the region, X: fixed capital.

Dataset 3. Source: Singh and Manghat [21], page 218.
Y: juice quantity per cane (grams), X: weight of cane (grams).The corresponding MSE values of the proposed estimators along with estimators used in the analysis are presented in Table 1 below.
Table 1 shows the mean square error value of the class of estimators, i.e., where i = 1, 2, 3, 4, and 5 is higher as compared to the , , , , , and .
The percentage relative efficiency is given by PRE =  where i = , , , , , , , , , , and .
Using the above formula, the PRE of all the estimators are calculated and presented in Table 2.
Table 2 shows that the PRE of where i = 1, 2, 3, 4, and 5. is higher as compared to , , , , , and .

#### 5. Efficiency Comparisons

The MSE’s of the proposed estimators are smaller under the following conditions:

MSE () < MSE () iff

On simplification, we get

Equation (22) holds for all i = 1, 2, 3, 4, and 5 as follows:(i) vs (ii) vs (iii) vs (iv) vs (v) vs (1)MSE () < MSE () iffOn simplification, we getEquation (29) holds for all i = 1, 2, 3, 4, 5 as follows:(vi) vs (vii) vs (viii) vs (ix) vs (x) vs (1)MSE () < MSE () iffOn simplification, we getEquation (36) holds for all i = 1, 2, 3, 4, and 5 as follows:(xi) vs