Abstract
In this paper, a ratio-exponential-log type general class of estimators is proposed in estimating the finite population mean using two auxiliary variables when population parameters of the auxiliary variables are known. From the proposed estimator, some special estimators are identified as members of the proposed general class of estimators. The mean square error (MSE) expressions are obtained up to the first order of approximation. This study finds that the proposed general class of estimators outperforms as compared to the conventional mean estimator, usual ratio estimators, exponential-ratio estimators, log-ratio type estimators, and many other competitor regression type estimators. Four real-life applications are used for efficiency comparison.
1. Introduction
In survey sampling, ratio, product, exponential-ratio, log-ratio, and regression type estimators are modified or constructed by many researchers to enhance the precision of the estimators under different sampling designs by using the auxiliary variables. These estimators are commonly used by taking the advantage of correlation coefficient between the study variable and the auxiliary variable(s). Some notable work by the authors includes Olkin [1]; Mohanty [2]; Abu-Dayyeh et al. [3]; Koyuncu and Kadilar [4]; Swain [5]; Lu and Yan [6]; Lu et al. [7]; Sanaullah et al. [8]; Lu [9]; Muneer et al. [10]; Shabbir and Gupta [11]; Akingbade and Okafor [12]; Shabbir et al. [13]; Shabbir et al. [14]; Bhushan et al. [15]; Lone et al. [16]; and Kumari and Thaur [17].
Consider a finite population of units. A sample of size units is drawn from a population by using simple random sampling without replacement (SRSWOR). Let and be the characteristics of the study variable and the auxiliary variables , respectively. Let , and , respectively, be the sample means corresponding to the population means , and . To obtain the bias and MSE expressions, we define the following error terms: , and , such that , , , , , , and where , , , , , , , , , , , , and .
2. Some Existing Estimators
Some existing estimators available in the literature are essential to be discussed here.
2.1. Sample Mean Estimator
The usual sample mean estimator and its variance are given as
2.2. Ratio Estimators
The usual ratio estimators when using single and two auxiliary variables are given by
The MSEs of ratio estimators to first order of approximation are given by
The ratio estimators are performing better than under certain conditions.
2.3. Exponential-Ratio Estimators
The usual exponential-ratio estimators when using single and two auxiliary variables are given by
The MSEs of exponential-ratio estimators to first order of approximation are given by
The exponential-ratio estimators are performing better than and under certain conditions.
2.4. Log-Ratio Estimators
Recently many log-type estimators have appeared in the literature in various forms when the logarithmic relationship between the study variable and the auxiliary variables exists.
The usual log-ratio estimators when using single and two auxiliary variables are given by
The MSEs of log-ratio estimators to first order of approximation are given by
The MSEs of log-ratio estimators are exactly equal to the MSEs of ratio estimators but their biases are different (not shown here).
2.5. Regression Estimators
The usual regression estimators when using single and two auxiliary variables are given bywhere and are the sample regression coefficients.
The MSEs of regression estimators are given by
These regression estimators are performing better than and under certain conditions.
2.6. Some More Regression Type Estimators
Mohanty [2] suggested the following regression-type estimator:
The MSE of is given by
Swain [5] introduced the following regression-type estimatorwhere is constant.
The minimum MSE of at optimum value of is given by
The unbiased regression estimator when using two auxiliary variables is given bywhere and are constants.
The minimum MSE of at optimum values of and is given bywhere is the multiple correlation coefficient.
3. Proposed General Class of Estimators
We propose a ratio-exponential-log type general class of estimators in estimating the finite population mean using two auxiliary variables when some parameters of the auxiliary variables are known. We also obtain different special estimators as members of the general class of estimators which are useful in different real-life situations. The proposed estimator is the combination of three special estimators including ratio, exponential-ratio, and log-ratio by using the linear transformation aswhere are constants, whose values are to be determined; and are scaler quantities; and , , , and . Here are the known population parameters of the auxiliary variables which may be coefficients of variation , coefficients of kurtosis and correlation coefficients .
Solving (25) in terms of errors to the first order of approximation, we havewhere
The bias of to the first order of approximation is given by
The MSE of to the first order of approximation is given by
Solving (29), we getwhere
Solving (30), the optimum values are given as
The minimum MSE of to the first order of approximation is given by
Some special estimators as members of the proposed general class of estimators are given by(i)Putting in (25), we get(ii)Putting in (25), we get(iii)Putting in (25), we get(iv)Putting in (25), we get(v)Putting in (25), we get(vi)Putting in (25), we get(vii)Putting in (25), we get(viii)Putting in (25), we get(i)Putting in (25), we get(x)Putting in (25), we get
Note: We can generate more sub-classes of the proposed general class of estimators by using different combinations.
4. Numerical Example
We use the following four real data sets for a numerical study. Population 1 (see [19]): Population 2 (see [18]): Population 3 (Punjab Development Statistics (2019)): This data is taken from Punjab development of statistics of 36 districts of Punjab, Pakistan during 2018. Population 4 (see [19]):
This data are based on 69 villages of Doraha development bloc of Punjab, India.
The results based on Populations 1–4 are given in Tables 1–11. We use the following expression to obtain the percent relative efficiency (PRE) aswhere .
In Table 1, we observed the following:(i)The ratio and log-ratio estimators in population 2, in populations 2 and 3, and in all four populations are performing poorly as compared to .(ii)The exponential-ratio estimator in Populations 2 and 3 is not performing good.(iii)Mohanty [2] regression estimator in populations 1–3 and Swain [5] estimator in Population 3 are not efficient as compared to.(iv)Among all the estimators above discussed, the performance of is the best.
5. Comparison of Estimators
Now we compare the proposed class of estimators with other existing estimators.
Condition 1. By (2) and (33), if where and .
Condition 2. By ((4)–(6)) or ((12)–(14)) and (33), if
Condition 3. By ((8)–(10)) and (33), if
Condition 4. By ((16)–(18)) and (33), if
Condition 5. By (20) and (38), if
Condition 6. By (22) and (38), if
Condition 7. By (32) and (38), if The proposed class of estimators will perform better when conditions 1–7 are satisfied.
6. Conclusion
In this study, we have proposed ratio-exponential-log type generalized class of estimators by combing a ratio, exponential-ratio, and log-ratio type estimators by using the linear transformation for finite population mean in simple random sampling. Expressions for the bias and MSE of proposed general class of estimators are obtained up to the first order of approximation. Four data sets are used for numerical study. Based on Tables 1–11, we observe that the proposed sub-classes of general estimators are performing well as compared to their competitor estimators. We have generated 10 sub-classes from the proposed general estimators with different combinations which all are efficient in different situation as compared to SRS. So, the proposed general class of estimators is preferable in further study.
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.