Abstract
This study suggests a new optimal family of exponential-type estimators for estimating population mean in stratified random sampling. These estimators are based on the traditional and nontraditional measures of auxiliary information. Expressions for the bias, mean square error, and minimum mean square error of the proposed estimators are derived up to first order of approximation. It is observed that proposed estimators perform better than the traditional estimators (unbiased, combined ratio, and combined regression) and other recent estimators. A real dataset is used to highlight the applicability of proposed estimators. In addition, a simulation study is carried out to assess the performance of new family as compared to other estimators.
1. Introduction
Nowadays, it is common practice to use the auxiliary/ancillary information to boost the efficiency of estimators in survey sampling. Most of the researchers only deal with the traditional information of auxiliary variable(s) such as standard deviation, coefficient of variation, coefficient of skewness, coefficient of kurtosis, and coefficient of correlation. Having edge of this traditional information, many authors have been trying to explore new optimal estimators and families of estimators for estimating population mean under stratified random sampling. Stratified random sampling has often proved needful in improving the precision of estimators over simple random sampling, for instance, see works of Kadilar and Cingi [1, 2], Koyuncu and Kadilar [3, 4], Singh and Vishwakarma [5–7], Shabbir and Gupta [8], Haq and Shabbir [9], Singh and Solanki [10], Yadav et al. [11], Solanki and Singh [10, 12], Javed et al. [13], and Javed and Irfan [14].
The motivation behind this article is to utilize the nontraditional information as well as the traditional information of the auxiliary variable to progress the estimation of population mean in stratified random sampling. This idea is initiated first time in this article under stratified random sampling.
Nontraditional information includes quartile deviation, midrange, interquartile range, quartile average, decile mean, tri-mean, Hodges–Lehmann estimator, and L-moments of an auxiliary variable. L-moments are determined by linear combinations of the expected values of the order statistics (for detail, check the works of Hosking [15] and Shahzad et al. [16]). Furthermore, efficiency of the estimators is uncertain in the occurrence of the extreme values in the dataset. Some of the above nontraditional measures such as decile mean, Hodges–Lehmann estimator, and tri-mean are robust measures. Utilizing these measures, we can well cope with the extreme values/outliers in the dataset. In addition, L-moments also are used to reduce the negative effect of outliers on the estimators.
Rest of the article is organized in the following way. Section 2 presents the useful notations. Section 3 gives comprehensive detail of existing families of estimators. Section 4 suggests a new optimal family of estimators for estimating population mean using traditional and nontraditional measures of auxiliary variable. Expressions for the bias, mean squared error (MSE) and minimum MSE of this family are derived up to first degree of approximation in the same section. A real dataset is used in Section 5 to check the potential of new estimators as compared to existing ones. In Section 6, the performance of suggested family is evaluated by carrying out a simulation study using the same dataset used in Section 5. Section 7 contains the final discussion.
2. Useful Notations
Let us consider a finite population of size , and it can be stratified into homogenous strata with stratum containing units subject to the restriction that A sample of size is drawn under simple random sampling without replacement (SRSWOR) from stratum such that Consider the pairs of observations , made from stratum for the study and auxiliary variables, respectively.
Furthermore, let be the population and sample means of , respectively, where are the population mean, sample mean, and the weight of stratum, respectively. Following the same lines, can be defined for the auxiliary variable .
To derive the expressions for the bias, mean square error (MSE), and minimum mean square error of the existing and proposed estimators, we consider the following relative error terms along with their expectations assuch that
From (2), we can write as below:where
Some other formulas for stratum, under stratified random sampling are listed below:where are the first, second, and third quartiles, respectively, is the minimum value, and is the maximum value of the data.
3. Some Existing Estimators/Classes of Estimators
This section gives a brief introduction of some well-known estimators/classes of estimators from the literature.
3.1. Usual Estimators
In stratified random sampling usual unbiased , combined ratio and combined regression estimators and their MSEs are detailed below:where .
Bahl and Tuteja [17] suggested ratio and product exponential-type estimators for population mean under stratified random sampling as
Average of (7) and (8) can be written as
3.2. Koyuncu and Kadilar [3]
To estimate the population mean under stratified random sampling, a family of ratio estimators was introduced by Koyuncu and Kadilar [3] as below:where and are suitable constants and and are either real numbers or functions of known parameters of the auxiliary variable such as coefficient of skewness, coefficient of kurtosis, coefficient of variation, and coefficient of correlation.
Up to the first order of approximation, the bias and MSE of are given by
3.3. Koyuncu and Kadilar [4]
Koyuncu and Kadilar [4] considered the ratio estimator of Gupta and Shabbir [18] and suggested an improved estimator defined as below:where and are suitably chosen weights.
Given below are the expressions, up to first degree of approximation, for the bias and MSE of , respectively:
The suitable weights of and are given by
Inserting the above weights of and in (14), we get the minimum MSE of as
3.4. Shabbir and Gupta [8]
Given below is a ratio-type estimator suggested by Shabbir and Gupta [8] in stratified random sampling:where and are the constants to be determined. Also, we consider that
Expressions for the bias and the MSE of , respectively, are given below:
The suitable weights of and are given as
Putting weights of and in (20), we have minimum MSE of as
3.5. Haq and Shabbir [9]
Haq and Shabbir [9] proposed two exponential ratio-type families of estimators detailed below:where are the suitable constants.
Given below are the expressions for bias and MSE of , respectively:where and is defined earlier.
The weights of are determined as below:
By substituting values of in (25) and in (26), we get minimum MSE of and , respectively:
3.6. Singh and Solanki [19]
Singh and Solanki [19] proposed a family of estimators as given below:where and are suitable scalars and are the constants to be determined to make the MSE minimum. Assuming different values of , and proposed family reduces to the ratio-type , product-type , and ratio-cum-product-type estimators.
For ratio-type, product-type, and ratio-cum-product-type estimators suitable values are , respectively.
For bias and MSE of , we consider the expressions given below:where
The weights of are determined as below:
Substituting the above weights in (31), we get the minimum MSE as given by
3.7. Solanki and Singh [10]
Given below is the class of estimators suggested by Solanki and Singh [10]:where are the feasible weights to be found such that the MSE is minimal. Here, ; and being the constants take values (0, 1, −1) for obtaining different estimators like(i)Ratio-type exponential estimators for (ii)Product-type exponential estimators for (iii)Ratio-ratio-type exponential estimators for (iv)Product-product-type exponential estimators for (v)Ratio-product-type exponential estimators for (vi)Product-ratio-type exponential estimators for
For bias and MSE of , we consider the expressions given below:where
The suitable weights of are as below:
Substituting these suitable weights in (37), we have the minimum MSE as given by
3.8. Solanki and Singh [12]
Recently, Solanki and Singh [12] defined an improved estimation given aswhere are the suitably chosen weights to get minimum MSE. , are real number to parameters related to auxiliary variate Here, , and being the constants take values (−1, 0, 1) for obtaining different classes of estimators :(i), for (ii), for (iii), for
For bias and MSE of , we consider the expressions given below:where :where
Given below are the weights of for minimizing the MSE:
Thus, the minimum MSE by putting the above values of in (43) is given by
4. Suggested Family of Estimators
Following the lines of Shabbir et al. [20], a generalized estimator for the estimation of population mean is proposed using some traditional and nontraditional measures of an auxiliary variable. For more details of these nontraditional measures, see the works of Hettmansperger and McKean [21], Wang et al. [22], and Irfan et al. [23–26]:where and are the suitably chosen weights and and are either real numbers or functions of known parameters of the auxiliary variable such as standard deviation , coefficient of skewness , coefficient of kurtosis , coefficient of variation , coefficient of correlation , quartile deviation , midrange , interquartile range , quartile average , tri-mean , and Hodge–Lehmann estimator .
Remark 1. Many more estimators can be generated by placing different available parameters of auxiliary variable in place of and . Some of them are presented in Table 1.
Using (1), the suggested class of estimators can be rewritten asAs defined earlier, .
Expanding the right-hand side of (48), up to first order of approximation and subtracting from both sides, we obtainUp to first order of approximation, the bias and the MSE are given byBy minimizing (51), suitable weights of and are obtained as below:Putting the optimal weights of and in (51), we have the minimum MSE given by
5. Application to a Dataset
To examine the performance of the proposed class of estimators, we considered a real data of Turkey (2007) used by Koyuncu and Kadilar [3] given in Table 2. In this data, let be the number of teachers (study variable) and be the number of students (auxiliary variable) which are recorded for primary and secondary schools at 6 regions for districts. A total sample of size is selected through Neyman allocation from 6 strata. (source: Ministry of Education, Republic of Turkey).
We have computed , for the population dataset given in Table 2 and are reported in Tables 3 and 4.
5.1. Important Findings
(i)The values of MSE of usual unbiased, combined ratio, and combined regression estimators in stratified random sampling are computed as given below:(ii)It is obvious from Table 3 that MSEs of Koyuncu and Kadilar [3] class of estimators round about the value of (iii)value of is less than and .(iv)It is observed from Tables 3 and 4 that the MSEs of Koyuncu and Kadilar [4] class of estimators , Shabbir and Gupta [8] class of estimators , Haq and Shabbir [9] classes of estimators , and Singh and Solanki [10, 12] classes of estimators , are less than the MSE of combined regression estimator .(v)Again, from Tables 3 and 4, it is valuable to mention that the proposed family of estimators have the least MSEs as compared to all other classes of estimators against different values of and .
6. Simulation Study
In this section, we carried out a simulation study using R statistical software to evaluate the behavior of proposed estimators in comparison with , Real population presented in Table 3 is used for the simulation study. Three different sample sizes are taken from this population on the basis of proportional allocation.
The following steps summarize the procedure of finding the average MSE of an estimator. Step 1: select a bivariate stratified sample of size using simple random sampling without replacement from the bivariate stratified normal population Step 2: use sample data from Step 1 to find the MSE of all the estimators under the study Step 3: repeat Step 1 and Step 2 10,000 times and obtain 10,000 values for MSEs Step 4: Average of 10,000 values obtained in Step 3 are the MSE of each estimator
6.1. Findings
MSEs of for different sample sizes are under Table 5.
MSEs of the other estimators under the study are presented in Tables 6–11, and the following important considerations are made from them.(i)It is quite obvious that the MSE values of , as compared to the MSEs of all other estimators are minimum under different sample sizes taken from the population. It proves that all the proposed estimators are more efficient.(ii)It is also shown that, by increasing the sample size selected from the population, the MSEs decrease.
7. Discussion
In this study, we proposed a new optimal family of estimators for estimating population mean under stratified random sampling. Bias, MSE, and minimum MSE of this family of estimators are derived up to first degree of approximation. The proposed family is compared with some well-known estimators/classes of estimators under stratified random sampling such as the works of Koyuncu and Kadilar [3, 4], Shabbir and Gupta [8], Haq and Shabbir [9], Singh and Solanki [19], and Solanki and Singh [10, 12]. It is numerically inferred that the proposed family behaves optimal as compared to other estimators. A simulation study is also carried out in support of efficient proposed estimators. So, to get more enhanced results in practice under stratified random sampling, our suggested family of estimators is recommended.
The possible extensions of this work are to estimate the (1) finite population mean using robust quantile regression and L-moments characteristics of an auxiliary information under stratified ranked set sampling, (2) finite population parameters including median, variance, and proportions using L-moments under different sampling designs, and (3) population mean in the presence of nonsampling errors using L-moments and calibration approach.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.