Abstract

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.

1. 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 exponential-ratio- 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 of units. A sample of size units is drawn from this population using simple random sampling without replacement (SRSWOR). Let and be the observed values of the study variable and the auxiliary variable , respectively. Let and , respectively, be the sample means and and be the corresponding population means. Let be the coefficient of variations and be the standard deviations of , respectively. Let be the correlation coefficient and be the covariance between the variables indicated by the subscripts. We define the following error terms to obtain the bias and MSE expressions: and such that , , , and , where

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 based on median and the two quartiles, and kurtosis was originated by Karl Pearson (1857–1936). Antonine Augustin Cournot was the first to use the term median in 1843. The midrange , i.e., was introduced by Robert K. Merton. Tukey [14] used the idea of Trimean (TM) , where are the first, second, and third quartiles of the auxiliary variable which were discussed by Tukey [15]. The term quartile deviation was used in “Proceeding of the Royal Statistical Society of London” in late 19th century. Hodge and Lehmann [16] used the measure : for estimation of location based on ranks.

2. Some Existing Estimators

We discuss the following mean estimators that exist in the literature:(i)The sample mean estimator is , and its variance is given by(ii)The usual ratio estimator proposed by Cochran [17] when the regression line Y on X passes through origin is given bywhere is the known population mean of the auxiliary variable X. The performance of ratio estimator is better as compared to the usual mean estimator when .The bias and MSE, respectively, of are given by(iii)Bahl and Tuteja [18] suggested the exponential-ratio-type estimator given byThe bias and MSE, respectively, of are given byThe exponential ratio estimator is superior to the usual mean estimator and ratio estimator if and , respectively.(iv)The regression or difference estimator [19, 20] is given bywhere is a constant.The minimum MSE of at is given byThe difference estimator always performs better than usual sample mean estimator, ratio estimator, and exponential ratio estimator if , and , respectively.(v)Singh et al.’s estimator [2] is given bywhere is a constant.The minimum MSE of at is given by(vi)Singh et al. [4] suggested the following estimator:where and are the functions of known population parameters of the auxiliary variable. Some members of the family of estimators are given in Table 1.The bias and MSE, respectively, of are given bywhere . We can get different values of by using different values of , i.e., , , , , , , , , , , , resulting in , respectively.(vii)Yadav and Kadilar [6] suggested the following estimator:where is a constant. Some members of the family of estimators are given in Table 1.The bias and minimum MSE, respectively, of are given bywhere and .The optimum value of is given by .(viii)Kadilar [9] suggested the following exponential-ratio-type estimator:where is a constant.The bias and MSE, respectively, of at are given by(ix)Grover and Kaur’s estimator [8] is given bywhere and are constants. Some members of the family of estimators are given in Table 1.The bias and minimum MSE, respectively, of at and are given bywhere , , , , and .(x)Recently, Irfan et al. [10] suggested the following exponential-type estimator:where and are constants. Some members of the family of estimators are given in Table 2.The bias and minimum MSE, respectively, of at and are given bywhere , , , , and .(xi)Shabbir et al. [7] suggested the following transformed exponential-ratio-difference-type estimator:where and and are constants. The members of the family of estimators are given in Table 2.The bias and minimum MSE, respectively, of , at and , are given bywhere and (xii)On the lines of Shabbir et al. [7], Irfan et al. [11] suggested an estimator, which is given bywhere and and are constants. The members of the family of estimators are given in Table 3.

The bias and minimum MSE, respectively, of , at and are given bywhere

3. Proposed Estimator

We propose a fairly simple class of exponential-ratio-type estimators using the conventional and nonconventional measures as given below:where and are constants and and are the known conventional and nonconventional measures of the auxiliary variable. Various members of the family of estimators are given in Table 3. The purpose of constructing this new class of estimators is to see its behavior when using both conventional and nonconventional measures.

Rewriting in terms of errors up to first order of approximation, we have

The bias of to first degree of approximation is given by

Taking square and then expectation in (35), the MSE of to first degree of approximation is given bywhere and .

The optimum values of and are and .

Substituting the optimum values in (37), we can get minimum MSE of which is given by

4. Comparison of Estimators

Now we compare the proposed class of estimators with other existing estimators discussed here.

Condition 1. By (1) and (37), if , where and .

Condition 2. By (4) and (37), if

Condition 3. By (7) and (38), if

Condition 4. By (9), (11), (20), and (38), if

Condition 5. By (14) and (38), if

Condition 6. By (17) and (38), if

Condition 7. By (23) and (38), ifwhere and .

Condition 8. By (26) and (38), ifwhere and .

Condition 9. By (29) and (38), ifwhere and , .

Condition 10. By (32) and (38), ifwhere and , .

5. Numerical Examples

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

5.1. 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 as but different covariance matrices as given below.

Population 1:

Population 2:

For each population, we consider a sample of sizes 50 and 100. The following steps are performed to carry out the simulation study.Step 1. Select a SRSWOR of size from a population of size .Step 2. Use a sample data from Step 1 to find the MSE values of all the estimators.Step 3. Steps 1 and 2 are repeated 10,000 times.Step 4. Obtain 10,000 values for MSEs.Step 5. Average of 10,000 values obtained in Step 4 represents the simulated MSE of each estimator.

The simulated MSEs based on Populations 1 and 2 for sample sizes 50 and 100 are given in Table 4.

From Table 4, we observed that as we increase the sample size, the MSE values decrease in both populations which are on expected lines. Irfan et al.’s estimator [11] becomes Shabbir et al.’s estimator [7] when , so MSE values are the same. The proposed estimator shows the least MSE values as compared to all other considered estimators.

5.2. Real Datasets

We use the following 7 real datasets for a numerical study.Population 1 (source: Singh and Chaudhary [21]):area under wheat crop in acres during 1974 in 34 villages.area under wheat crop in acres during 1971 in 34 villages.The summary statistics are and .Population 2 (source: Cochran [19]):number of inhabitants (in 1000’s) in 1930.number of inhabitants (in 1000’s) in 1920.The summary statistics are and .Population 3 (source: Singh and Mangat [22]):number of tube wells.net irrigated area in hectares for 69 villages.The summary statistics are and .Population 4 (source: Singh and Mangat [22]):average duration of sleep in hours.age of a person.The summary statistics are and .Population 5 (source: Gujarati [23, p. 433]):average miles per gallons.top speed miles per hour of 81 cars.The summary statistics are and .Population 6 (source: Singh and Chaudhary [21]):area under wheat crop in acres during 1974 in 34 villages.area under wheat crop in acres during 1973 in 34 villages.The summary statistics are and .Population 7 (source: Gujarati [23, p. 433]):average miles per gallons.cubic feet of cab space of 81 cars.The summary statistics are and .

The results based on Populations 1–7 are given in Tables 516, where we use the following expression to obtain the percent relative efficiency (PRE):where .

The results based on 7 real datasets are given in Tables 516.

In Tables 516, 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 , and are equally efficient. Also, some estimators show very poor performances in Populations 4, 5, and 7 because of negative correlations. In Tables 516, all of the estimators have very large PRE for Population 2 due to the highest value of Also, in all cases, the proposed estimator shows the best performance.

6. 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. The 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 616. Based on the results in Tables 416, the proposed class of estimators performs substantially better than all other estimators considered here. In Table 5, the estimators perform very poorly for Populations 4, 5, and 7 because of negative correlation. Estimators in Tables 616 also performed poorly for Populations 4, 5, and 7 due to negative correlation. We observed that the efficiency of the estimators (see Tables 616) in some situations does not increase much as compared to other estimators. The 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

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.