Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2021 / Article
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Robust Estimation Methods in the Presence of Extreme Observations

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Research Article | Open Access

Volume 2021 |Article ID 3741620 | https://doi.org/10.1155/2021/3741620

Javid Shabbir, Sat Gupta, Ronald Onyango, "On Using the Conventional and Nonconventional Measures of the Auxiliary Variable for Mean Estimation", Mathematical Problems in Engineering, vol. 2021, Article ID 3741620, 13 pages, 2021. https://doi.org/10.1155/2021/3741620

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

Academic Editor: Ishfaq Ahmad
Received16 May 2021
Accepted12 Jun 2021
Published29 Jun 2021

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.


Singh et al. [4]Yadav and Kadilar [6]Grover and Kaur [8]

10
1

and

Irfan et al. [10]Shabbir et al. [7]

10
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