Confidence intervals for the weighted coefficients of variation of two-parameter exponential distributions

This paper proposes new confidence intervals for the weighted coefficients of variation (CV) of two-parameter exponential distributions based on the adjusted method of variance estimates recovery method (adjusted MOVER). This is then compared with the generalized confidence interval method (GCI) and the large sample method. The performance of these confidence intervals in terms of coverage probabilities and average lengths were evaluated via a Monte Carlo simulation. Simulation studies showed that the GCI should be considered as an alternative to the confidence interval estimation for the weighted CV of two-parameter exponential distributions. However, the adjusted MOVER confidence interval (CI AM2 ) can be used to estimate the weighted CV when the coefficient of variation is a positive value. The proposed confidence intervals are illustrated using a real example. Subjects: Science; Mathematics & Statistics; Statistics & Probability; Statistics


Introduction
In probability and statistics, the two-parameter exponential distribution is used to represent the time to failure in many applications, such as lifetime data, survival, and reliability analysis (Hahn & Meeker, 1991). This distribution is widely used in many fields and confidence interval estimation of its parameter is importance. The confidence interval provides information respecting the population ABOUT

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The problem of estimating the parameter of two-parameter exponential distribution has been studied continuously. In practice, the data are collected at different settings. This study has provided two methods and proposed a novel method for confidence interval estimation for the weighted coefficients of variation of twoparameter exponential distributions based on the adjusted method of variance estimates recovery method (adjusted MOVER), then compared with the generalized confidence interval method (GCI), and the large sample method. It is concluded that the GCI should be considered as an alternative to the confidence interval estimation for the weighted coefficients of variation, whereas the adjusted MOVER confidence interval CI AM2 can be used when the coefficient of variation is a positive.
value of the quantity much more than the point estimate (Casella & Berger, 2002). Therefore, many studies have examined confidence interval estimation for the parameters in this distribution. For example, Chiou (1997) presented a method for confidence interval estimation of scale parameters following a pre-test for two exponential distributions. Roy and Mathew (2005) proposed the generalized confidence interval approach to construct an exact lower confidence limit for the reliability function of a two-parameter exponential distribution. Li and Zhang (2010) considered the problem of estimation of asymptotic confidence interval for the ratio of means of two-parameter exponential distributions. Kharrati-Kopaei, Malekzadeh, and Sadooghi-Alvandi (2013) presented simultaneous fiducial generalized confidence intervals for the successive differences of two-parameter exponential location parameters in three populations when both scale parameters and sample sizes are possibly unequal. Singh and Singh (2013) proposed simultaneous confidence intervals of ordered pairwise differences of exponential location parameters under heteroscedasticity. Li, Song, and Shi (2015) proposed a parametric bootstrap method to construct simultaneous confidence intervals for all pairwise differences of means from several two-parameter exponential distributions.
Coefficient of variation is the ratio of the standard deviation to the mean (Kelley, 2007). It is a measure of relative variability. The coefficient of variation is often used to compare two distributions measured on different units. The coefficient of variation has been used in many fields, such as science, medicine, economics, and life insurance. For example, it is used to analyze the cycle variation in hydrogen-fueled engines with direct injection (Kim, Lee, & Choi, 2005). It is also used to measure the variation in socioeconomic status and the prevalence of smoking in tobacco control environments (Gulhar, Kibria, Albatineh, & Ahmed, 2012). In medical study, the coefficient of variation is used to measure precision within and between laboratories (Tian, 2005). In diet study, the coefficient of variation is used to compare the variability in the ratio of total/HDL cholesterol with the variability in vessel diameter change because the ratio of total/HDL cholesterol and the vessel diameter change are measured in different units (Tian, 2005). Several researchers have studied focusing on confidence interval for the coefficient of variation, for example, Vangel (1996) presented confidence intervals for a normal coefficient of variation. Tian (2005) developed the procedures for confidence interval estimation and hypothesis testing for the common coefficient of variation of normal distributions. Wong and Wu (2002) proposed confidence intervals for the coefficient of variation of normal and nonnormal models. Mahmoudvand and Hassani (2009) proposed two new confidence intervals for the coefficient of variation in a normal distribution. Curto and Pinto (2009) proposed statistical tests for two coefficients of variation comparison in non-iid case. Banik and Kibria (2011) reviewed several interval estimators proposed by different researcher at different times for estimating the population coefficient of variation and compared with bootstrap interval estimators when data are generated from various distributions. Niwitpong (2013) presented the confidence intervals for coefficient of variation of log-normal distribution with restricted parameter space.
In practical applications, the independent samples are collected from different settings. Therefore, inference procedures regarding several coefficients of variation are of interest. Several researchers proposed statistical tests to test the equality of two or more coefficients of variation, i.e. see Ahmed (1995), Gupta and Ma (1996), Fung and Tsang (1998), Curto and Pinto (2009), and Gokpinar and Gokpinar (2015).  proposed confidence intervals for the single coefficient of variation and the difference of coefficients of variation in the two-parameter exponential distributions.  presented confidence intervals for the ratio of coefficients of variation in the two-parameter exponential distributions. Tian (2005) noted that confidence interval estimation or hypothesis testing about the common population coefficient of variation from several sample is need in many circumstances. Combine the data of several independent samples is used in clinical trials and social and behavioral sciences. Collecting independent sample from different populations with the common coefficient of variation but possibly with different variances, then the problem of interest is to estimate or construct a confidence interval for the common coefficient of variation. This problem arises in situations where different instruments, different methods, or different laboratories are used to measure the products to estimate the average quality. Tian (2005) proposed the generalized variable approach to make inferences about the common coefficient of variation based on several independent normal samples. Ng (2014) presented the generalized variable approach for making inferences about the common coefficient of variation based on several independent log-normal samples. To our knowledge, no paper exists for weighted coefficients of variation in k two-parameter exponential populations. Then, inference procedures are of practical and theoretical importance, to develop procedures for confidence interval estimation for the weighted coefficients of variation of two-parameter distributions. Hence, this paper will fill this gap by developing novel methods for confidence interval for the weighted coefficients of variation in several two-parameter exponential populations. We applied the results of  for the weighted coefficients of variation in two or more populations. Moreover, this paper searches for a confidence interval for weighted coefficients of variation of several twoparameter exponential populations that is easy to use in practice.
The samples are collected from several independent two-parameter exponential populations which are of interest. Therefore, the goal of this paper is to provide two methods and propose a novel method for confidence interval estimation of weighted coefficients of variation of two-parameter exponential derived from several independent samples. The first method was constructed based on the concept of generalized confidence interval method (GCI). Weerahandi (1993) introduced the GCI which could successfully construct the confidence interval for common parameters; for example, see Krishnamoorthy and Lu (2003), Tian (2005), Tian and Wu (2007), and Ye, Ma, and Wang (2010). The second method was constructed according to the large sample method which was based on central limit theorem (CLT). Tian and Wu (2007) presented the GCI and the large sample method to construct confidence intervals for the common mean of several log-normal populations. The third method, the proposed method, was constructed based on the adjusted method of variance estimates recovery method (adjusted MOVER). The adjusted MOVER method was motivated and extended based on the method of variance estimates recovery method (MOVER), for example see, Zou and Donner (2008) and Zou, Taleban, and Hao (2009), and was inspired by the score interval method proposed by Bartlett (1953). Several researchers have successfully used the MOVER method to construct the confidence interval; see i.e. Zou and Donner (2008), Zou et al. (2009), Donner and Zou (2010), and . From our knowledge, there are no proposed methods for the weighted coefficients of variation of several two-parameter exponential populations.
The organization of this paper is as follows. Section 2 describes the theory and computational procedures to construct the confidence intervals. Section 3 demonstrates the simulation results and illustrates the proposed methods with a real example. Section 4 summarizes this paper.

The generalized confidence interval method (GCI)
Let X = X 1 , X 2 , … , X n be a random variable follows a two-parameter exponential distribution with probability density function where is a scale parameter and is a location parameter.
The mean and variance of X are The maximum likelihood estimators of parameters and are Let X i , i = 1, 2, … , k be random samples from two-parameter exponential distributions. Let be the coefficient of variation of X i . This paper is interested in constructing confidence intervals for the weighted coefficients of variation, based on Graybill and Deal (1959), defined as follows: where ̃i is an unbiased estimator based on the i-th sample; see Lin and Lee (2005).
Let X i and X (1)i denote the sample mean and the smallest sample for data X ij , i = 1, 2, … , k; j = 1, 2, … , n i . And let x i and x (1)i denote the observed sample mean and the smallest observed sample, respectively.
The maximum likelihood estimator of i is According to , the expectation and variance of ̂i are defined as follows: and Thus, ̂i is a biased estimator of i . To use the pooled estimators of Graybill and Deal (1959), an unbiased estimator of i is needed. The unbiased estimator is defined by The expectation of ̃i is The variance of ̃i is Following Weerahandi (1993): let X = X 1 , X 2 , … , X n be a random sample from a distribution F X (x; , ), where is a scalar parameter of interest and is a nuisance parameter. Let x = x 1 , x 2 , … , x n be an observed sample. A generalized confidence interval for is computed using the percentiles of a generalized pivotal quantity R(X;x, , ) which is a function of X, x, and if the following two conditions are satisfied: (i) For a given x, the distribution of R(X; x, , ) is free of all unknown parameters.
(ii) The observed value of R(X; x, , ), X = x, is the parameter of interest.
As in Lawless (1982), then and (3) where 2 2n i −2 denotes a chi-square distribution with degrees of freedom 2n i − 2 and 2 2 denotes a chi-square distribution with degrees of freedom 2. Then and The generalized pivotal quantity for i is defined as follows: The generalized pivotal quantity for i is defined as follows: According to , the generalized pivotal quantity for i is According to Ye et al. (2010), the generalized pivotal quantity for the weighted coefficients of variation is a weighted average of the generalized pivot R i based on k individual samples defined as follows: where (from Equation (4)) The generalized confidence interval for the weighted coefficients of variation can be constructed from R . Therefore, the 100(1 − )% two-sided confidence interval for the weighted coefficients of variation based on GCI is where R ∕2 and R 1 − ∕2 denote the ∕2-th and 1 − ∕2-th quantiles of R , respectively.

The large sample method
The large sample estimate of the coefficient of variation is a pooled estimated unbiased estimator of the coefficient of variation, based on Graybill and Deal (1959), defined as follows: where and it is easy to see that Var ̃i is Therefore, the 100(1 − )% two-sided confidence interval for the weighted coefficients of variation based on the large sample method is where z 1− ∕2 denotes the 1 − ∕2-th quantile of the standard normal distribution.

The adjusted method of variance estimates recovery method (adjusted MOVER)
For two parameters case, the method of variance estimates recovery method (MOVER) was introduced by Zou and Donner (2008) and Zou et al. (2009). Let 1 and 2 be the parameters of interest.
The MOVER method is used to construct a 100(1 − )% two-sided confidence interval of 1 + 2 .
Using the central limit theorem and the assumption of independence between the point estimates ̃1 and ̃2 , the lower limit L is defined as follows: where z ∕2 denotes the ∕2-th quantile of the standard normal distribution.
The l 1 , u 1 and l 2 , u 2 contain the parameter values for 1 and 2 , respectively. The lower limit L must be closer to l 1 + l 2 than to ̃1 +̃2. The variance estimate for ̃1 and ̃2 at i = l i is

Substituting back into Equation (15), then
By performing similar steps with this idea, the upper limit U must be closer to u 1 + u 2 , which gives Consider k parameters case, let 1 , 2 , … , k be the parameters of interest. The MOVER method is motivated to construct a 100(1 − )% two-sided confidence interval for 1 + 2 + ⋯ + k . Using the central limit theorem and the assumption of independence between the point estimates ̃1 ,̃2, … ,̃k, the lower limit L is defined as follows: where z ∕2 denotes the ∕2-th quantile of the standard normal distribution.
The l 1 , u 1 , l 2 , u 2 , … , l k , u k contain the parameter values for 1 , 2 , … , k , respectively. The lower limit L must be closer to l 1 + l 2 + ⋯ + l k than to ̃1 +̃2 + ⋯ +̃k. The lower limit L for 1 + 2 + … + k is and similarly, the upper limit (14) The adjusted MOVER method was motivated based on concepts of the large sample method in Equations (13)-(14) and MOVER method in Equations (15)-(19). According to Graybill and Deal (1959), the weighted coefficients of variation is weighted average of the coefficient of variation ̃i based on k individual samples is defined as where the variance estimate for ̃i at i = l i and i = u i is the average variance between these two variances and given by The lower limit L and upper limit U for the weighted coefficients of variation are and Therefore, the 100(1 − )% two-sided confidence interval for the weighted coefficients of variation based on adjusted MOVER method is where z ∕2 and z 1− ∕2 denote the ∕2-th and 1 − ∕2-th quantiles of the standard normal distribution, respectively.
The lower limit l 1i and upper limit u 1i for coefficient of variation i = 1i , are given by where (20) Moreover, using the normal approximation, the pivotal statistic is where Therefore, the confidence interval for coefficient of variation i is given by where Therefore, the 100(1 − )% two-sided confidence intervals for the weighted coefficients of variation based on adjusted MOVER method are

Comparative analysis
In this section, the performance of four confidence intervals in terms of coverage probabilities and average lengths are compared via a Monte Carlo simulation. The generalized confidence interval is defined as CI GCI , the large sample confidence interval is defined as CI LS , and two adjusted MOVER confidence intervals are defined as CI AM1 and CI AM2 . A confidence interval, with the values of coverage probability at least or close to the nominal confidence level 1 − and also has the shortest average length, was chosen.
Tables 1-6 presents the coverage probabilities and average lengths of confidence intervals for the weighted coefficients of variation of two-parameter exponential distributions for 2, 4, and 6 sample cases, respectively. The results show that the adjusted MOVER confidence interval CI AM2 performs as well as the generalized confidence interval CI GCI for large sample size, i.e. n ≥ 100 and is a positive value. For a negative value of , the generalized confidence interval CI GCI performs the best confidence interval compared with the other confidence intervals. Hence, the generalized confidence interval CI GCI and the adjusted MOVER confidence interval CI AM2 can be used for estimating the weighted coefficients of variation of two-parameter exponential distributions when is a positive value only, otherwise the generalized confidence interval CI GCI will be chosen. Note that, from simulation results, the large sample confidence interval CI LS performs as well as the adjusted MOVER confidence interval CI AM2 for every case.
The following algorithm can be used to estimate the coverage probability and average length.