On Using the Conventional and Nonconventional Measures of the Auxiliary Variable for Mean Estimation

In this paper, we propose an improved new class of exponential-ratio-type estimators for estimating the finite population mean using the conventional and the nonconventional measures of the auxiliary variable. Expressions for the bias and MSE are obtained under large sample approximation. Both simulation and numerical studies are conducted to validate the theoretical findings. Use of the conventional and the nonconventional measures of the auxiliary variable is very common in survey research, but we observe that this does not add much value in many of the estimators except for our proposed class of estimators.


Introduction
Many research papers have appeared in the literature where authors have used the conventional and the nonconventional measures of the auxiliary variable to enhance the efficiency of estimators. Recently, Gulzar et al. [1] used the nonconventional measures for population variance. Many other researchers have contributed to this area. Singh et al. [2] suggested a class of linear combinations of exponentialratio-and product-type estimators for mean estimation. Gupta and Shabbir [3] introduced a class of ratio-in-difference-type estimators using the conventional measures for the population mean. Singh et al. [4] presented an improved family of exponential-ratio-type estimator in simple random sampling for population mean. Haq and Shabbir [5] introduced an improved family of ratio-type estimators in simple and stratified sampling. Yadav and Kadilar [6] proposed an exponential family of ratio-type estimators by using conventional measures for estimating the population mean. Shabbir et al. [7] presented a new family of estimators for finite population mean in simple random sampling. Grover and Kaur [8] suggested a generalized class of exponential-ratio-type exponential estimators using the conventional measures for mean estimation. Kadilar [9] discussed a new exponential-type estimator for mean estimation. Irfan et al. [10] suggested a generalized ratio-exponential-type estimator using the conventional measures. Also, Irfan et al. [11] used both conventional and nonconventional measures in their proposed estimator. Ali et al. [12] used the robust-regression-type estimators for mean estimation of sensitive variables, and Shahzad et al. [13] suggested the L-moments based on calibration variance estimators in their recent work.
In our study, we propose a new generalized class of exponential-ratio-type estimators by using the conventional and the nonconventional measures of the auxiliary variable and compare our proposed estimator with several existing estimators.
Consider a finite population Λ � Λ 1 , Λ 2 , . . . , Λ N of N units. A sample of size n units is drawn from this population using simple random sampling without replacement (SRSWOR). Let y i and x i be the observed values of the study variable (Y) and the auxiliary variable (X), respectively. Let y � n − 1 n i�1 y i and x � n − 1 n i�1 x i , respectively, be the sample means and Y � N − 1 N i�1 y i and X � N − 1 N i�1 x i be the corresponding population means.
Let (C y � Y − 1 S y , C x � X − 1 S x ) be the coefficient of variations and(S y � 2 ) be the standard deviations of (Y, X), respectively. Let ρ yx � (S y S x ) − 1 S yx be the correlation coefficient and S yx � (N − 1) − 1 N i�1 (y i − Y)(x i − X) be the covariance between the variables indicated by the subscripts. We define the following error terms to obtain the bias and MSE expressions: Θ 0 � (y/Y) − 1 and Now we discuss some of the conventional and nonconventional measures of the auxiliary variable which are used in our study. Arthur Lyon Bowley (1869-1957), a British Statistician, introduced a term skewness (β 1 (x)) based on median and the two quartiles, and kurtosis (β 2 (x)) was originated by Karl Pearson (1857-1936). Antonine Augustin Cournot was the first to use the term median (M D ) in 1843. e midrange (M R ), i.e., (M R � ((X (1) + X (N) )/2)) was introduced by Robert K. Merton. Tukey [14] used the idea of Trimean (TM) are the first, second, and third quartiles of the auxiliary variable which were discussed by Tukey [15]. e term quartile deviation (Q D � ((Q 3 (x) − Q 1 (x))/2)) was used in "Proceeding of the Royal Statistical Society of London" in late 19 th century. Hodge and Lehmann [16] used the measure (H L ): (H L � median((X j + X k )/2)), 1 ≤ j ≤ k ≤ N, for estimation of location based on ranks.

Some Existing Estimators
We discuss the following mean estimators that exist in the literature: (i) e sample mean estimator is Y (0) � y, and its variance is given by (ii) e usual ratio estimator proposed by Cochran [17] when the regression line Y on X passes through origin is given by where X is the known population mean of the auxiliary variable X. e performance of ratio estimator is better as compared to the usual mean estimator y when ρ yx > (0.5C x /C y ).
e bias and MSE, respectively, of Y (R) are given by (iii) Bahl and Tuteja [18] suggested the exponentialratio-type estimator given by e bias and MSE, respectively, of Y (EP) are given by e exponential ratio estimator Y (EP) is superior to the usual mean estimator Y (0) and ratio estimator (iv) e regression or difference estimator [19,20] is given by where w is a constant.
e difference estimator always performs better than usual sample mean estimator, ratio estimator, and exponential ratio estimator if ρ 2 yx > 0, (C x − C y ρ yx ) 2 > 0, and (0.5C x − C y ρ yx ) 2 > 0, respectively. (v) Singh et al.'s estimator [2] is given by where s is a constant. e minimum MSE of Y (vi) Singh et al. [4] suggested the following estimator: where u and v are the functions of known population parameters of the auxiliary variable. Some members of the family of estimators Y (S 2 ) i (i � 1, 2, . . . , 12) are given in Table 1. are given by where φ i � (uX/(uX + v))(i � 1, 2, . . . , 12). We can get different values of φ i (i � 1, 2, . . . , 12) by using different values of (u, v), i.e., (u � 1, v � 0), respectively. (vii) Yadav and Kadilar [6] suggested the following estimator: where k is a constant. Some members of the family of estimators Y where . e optimum value of k is given by (viii) Kadilar [9] suggested the following exponentialratio-type estimator: where m is a constant. e bias and MSE, respectively, of Y (ix) Grover and Kaur's estimator [8] is given by  [6], and Grover and Kaur [8].

Mathematical Problems in Engineering 3
where g 1 and g 2 are constants. Some members of the family of estimators Y (GK) i (i � 1, 2, . . . , 12) are given in Table 1. e bias and minimum MSE, respectively, of Y where (x) Recently, Irfan et al. [10] suggested the following exponential-type estimator: where I 1a and I 2a are constants. Some members of the family of estimators Y Table 2. e bias and minimum MSE, respectively, of Y where [7] suggested the following transformed exponential-ratio-difference-type estimator: where  Table 2.
e bias and minimum MSE, respectively, of On the lines of Shabbir et al. [7], Irfan et al. [11] suggested an estimator, which is given by where Ω � (y/2) exp(X − x/(X + x)) + exp((x − X)/x + X)} and I 1b and I 2b are constants. e members of the family of estimators Y i (i � 1, 2, . . . , 12) are given in Table 3. e bias and minimum MSE, respectively, of Y where

Proposed Estimator
We propose a fairly simple class of exponential-ratio-type estimators using the conventional and nonconventional measures as given below: where T 1 and T 2 are constants and u and v are the known conventional and nonconventional measures of the auxiliary variable. Various members of the family of estimators Y (P) i (i � 1, 2, . . . , 12) are given in Table 3. e purpose of constructing this new class of estimators is to see its behavior when using both conventional and nonconventional measures.
Rewriting Y (P) in terms of errors up to first order of approximation, we have Mathematical Problems in Engineering to first degree of approximation is given by Taking square and then expectation in (35), the MSE of Y (P) to first degree of approximation is given by where Substituting the optimum values in (37), we can get which is given by

Numerical Examples
Both simulation and numerical studies are conducted to observe the performances of different estimators.

Simulation Study.
In this section, a simulation study is conducted to assess the performances of all estimators considered here. We consider two finite populations of size 1000 generated from a bivariate normal distribution with the same theoretical means of [Y, X] as μ � [5,5] but different covariance matrices as given below. Population 1: Σ � 9 1.9 1.9 4 . (48) Population 2: For each population, we consider a sample of sizes 50 and 100. e following steps are performed to carry out the simulation study.
Step 1. Select a SRSWOR of size n from a population of size N .
Step 2. Use a sample data from Step 1 to find the MSE values of all the estimators.
Step 5. Average of 10,000 values obtained in Step 4 represents the simulated MSE of each estimator. e simulated MSEs based on Populations 1 and 2 for sample sizes 50 and 100 are given in Table 4.
From Table 4 shows the least MSE values as compared to all other considered estimators.

Real Datasets.
We use the following 7 real datasets for a numerical study.                Tables 5-16, where we use the following expression to obtain the percent relative efficiency (PRE): where j � 0, R, EP, D(� S 1 , K), S 2 , GK, YK, I a , SH, I b , P. e results based on 7 real datasets are given in Tables 5-16.
In Tables 5-16, PRE values are given based on summary statistics of seven real datasets to observe the performances of all estimators. One can see that the estimators [Y  Tables 5-16, all of the estimators have very large PRE for Population 2 due to the highest value of ρ yx � 0.9817. Also, in all cases, the proposed estimator shows the best performance.

Conclusion
In this study, we have proposed a general class of exponential-ratio-type estimators for finite population mean in simple random sampling using the conventional and the nonconventional measures. Expressions for biases and MSEs are obtained up to first order of approximation. Two datasets are used for simulation study and seven real datasets are used for efficiency comparisons. e simulated results given in Table 4 show that the proposed estimator has the least MSE values as compared to other competitive estimators. Estimators in Table 5 do not use the conventional and nonconventional measures, but such measures are used in Tables 6-16. Based on the results in Tables 4-16, the proposed class of estimators performs substantially better than all other estimators considered here. In Table 5, the estimators Y (j) , (j � R, EP, S 2 , YK) perform very poorly for Populations 4, 5, and 7 because of negative correlation.
Estimators Y (j) , (j � S 2 , YK) in Tables 6-16 also performed poorly for Populations 4, 5, and 7 due to negative correlation. We observed that the efficiency of the estimators Y (j) , (j � S 2 , GK, YK, I a , I b ) (see  in some situations does not increase much as compared to other estimators.
e PRE of Irfan et al.'s estimator [11] was undefined in Table 15 under Population 6. We conclude that use of conventional and nonconventional measures does not play a very major role in increasing the efficiency of existing mean estimators except in some cases. However, we observed a significant gain for our proposed estimator with both the conventional and the nonconventional measures.

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.