Measuring Performance of Ratio-Exponential-Log Type General Class of Estimators Using Two Auxiliary Variables

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.

Consider a finite population Γ � Γ 1 , Γ 2 , . . . , Γ N of Nunits. A sample of size n units is drawn from a population by using simple random sampling without replacement (SRSWOR). Let y i and (x i , z i ) be the characteristics of the study variable (Y) and the auxiliary variables (X, Z), respectively. Let y � n − 1 n i�1 y i , and (x � n −1 n i�1 x i , z � n −1 n i�1 z i ), respectively, be the sample means corresponding to the population means Y � N −1 N i�1 y i , and To obtain the bias and MSE expressions, we define the following error terms: Ξ 0 � (y/Y) − 1, and Ξ 1 � (x/X) − 1,

Some Existing Estimators
Some existing estimators available in the literature are essential to be discussed here.

Sample Mean Estimator.
e usual sample mean estimator and its variance are given as Var

Ratio Estimators.
e usual ratio estimators when using single and two auxiliary variables are given by e MSEs of ratio estimators Y (i) (R) (i � 1, 2, 3) to first order of approximation are given by e ratio estimators Y (i) (R) (i � 1, 2, 3) are performing better than Y (0) under certain conditions.

Exponential-Ratio Estimators.
e usual exponentialratio estimators when using single and two auxiliary variables are given by e MSEs of exponential-ratio estimators Y to first order of approximation are given by e exponential-ratio estimators Y

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. e usual log-ratio estimators when using single and two auxiliary variables are given by e MSEs of log-ratio estimators Y (i) (log) (i � 1, 2, 3) to first order of approximation are given by e MSEs of log-ratio estimators Y

Regression Estimators.
e usual regression estimators when using single and two auxiliary variables are given by where b yx � (s yx /s 2 x ) and b yz � (s yz /s 2 z ) are the sample regression coefficients. e MSEs of regression estimators Y (i) (Reg) (i � 1, 2, 3) are given by  [2] suggested the following regression-type estimator: (20) Swain [5] introduced the following regression-type estimator e unbiased regression estimator when using two auxiliary variables is given by where d 1 and d 2 are constants. where is the multiple correlation coefficient.

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. e proposed estimator is the combination of three special estimators including ratio, exponential-ratio, and log-ratio by using the linear transformation as where M i (i � 1, 2) are constants, whose values are to be determined; α i (i � 1, 2, 3) and c i (i � 1, 2, 3) are scaler quantities; and Here a, b, c, d are the known population parameters of the auxiliary variables which may be coefficients of variation (C x , C z ), coefficients of kurtosis (β 2x , β 2z ) and correlation coefficients (ρ yx , ρ yz ). Solving (25) in terms of errors to the first order of approximation, we have where Mathematical Problems in Engineering , to the first order of approximation is given by to the first order of approximation is given by Solving (29), we get where

Mathematical Problems in Engineering
Solving (30), the optimum values M i (i � 1, 2) are given as to the first order of approximation is given by (33) Some special estimators as members of the proposed general class of estimators are given by Mathematical Problems in Engineering Note: We can generate more sub-classes of the proposed general class of estimators by using different combinations.

Numerical Example
We use the following four real data sets for a numerical study.
Population 1 (see [19]):    Tables 1-11. We use the following expression to obtain the percent relative efficiency (PRE) as In Table 1, we observed the following: (i) e ratio and log-ratio estimators (Y (log) ) in population 2, (Y

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
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.