Confidence Intervals for the Coefficient of Variation in a Normal Distribution with a Known Population Mean

This paper presents three confidence intervals for the coefficient of variation in a normal distribution with a known population mean. One of the proposed confidence intervals is based on the normal approximation. The other proposed confidence intervals are the shortest-length confidence interval and the equal-tailed confidence interval. AMonte Carlo simulation studywas conducted to compare the performance of the proposed confidence intervals with the existing confidence intervals. Simulation results have shown that all three proposed confidence intervals perform well in terms of coverage probability and expected length.


Introduction
The coefficient of variation of a distribution is a dimensionless number that quantifies the degree of variability relative to the mean [1].It is a statistical measure for comparing the dispersion of several variables obtained by different units.The population coefficient of variation is defined as a ratio of the population standard deviation () to the population mean () given by  = /.The typical sample estimate of  is given as where  is the sample standard deviation, the square root of the unbiased estimator of population variance, and  is the sample mean.The coefficient of variation has been widely used in many areas such as science, medicine, engineering, economics, and others.For example, the coefficient of variation has also been employed by Ahn [2] to analyze the uncertainty of fault trees.Gong and Li [3] assessed the strength of ceramics by using the coefficient of variation.Faber and Korn [4] applied the coefficient of variation as a way of including a measure of variation in the mean synaptic response of the central nervous system.The coefficient of variation has also been used to assess the homogeneity of bone test samples to help determine the effect of external treatments on the properties of bones [5].Billings et al. [6] used the coefficient of variation to study the impact of socioeconomic status on hospital use in New York City.In finance and actuarial science, the coefficient of variation can be used as a measure of relative risk and a test of the equality of the coefficients of variation for two stocks [7].Furthermore, Pyne et al. [8] studied the variability of the competitive performance of Olympic swimmers by using the coefficient of variation.
Although the point estimator of the population coefficient of variation shown in (1) can be a useful statistical measure, its confidence interval is more useful than the point estimator.A confidence interval provides much more information about the population characteristic of interest than does a point estimate (e.g., Smithson [9], Thompson [10], and Steiger [11]).There are several approaches available for constructing the confidence interval for .McKay [12] proposed a confidence interval for  based on the chi-square distribution; this confidence interval works well when  < 0.33 [13][14][15][16][17]. Later, Vangel [18] proposed a new confidence interval for , which is called a modified McKay's confidence interval.His confidence interval is based on an analysis of the distribution of a class of approximate pivotal quantities for the normal coefficient of variation.In addition, modified McKay's confidence interval is closely related to McKay's confidence interval but it is usually more accurate and nearly exact under normality.Panichkitkosolkul [19] modified McKay's confidence interval by replacing the sample coefficient of variation with the maximum likelihood estimator for a normal distribution.Sharma and Krishna [20] introduced the asymptotic distribution and confidence interval of the reciprocal of the coefficient of variation which does not require any assumptions about the population distribution to be made.Miller [21] discussed the approximate distribution of κ and proposed the approximate confidence interval for  in the case of a normal distribution.The performance of many confidence intervals for  obtained by McKay's, Miller's, and Sharma-Krishna's methods was compared under the same simulation conditions by Ng [22].
Mahmoudvand and Hassani [23] proposed an approximately unbiased estimator for  in a normal distribution and also used this estimator for constructing two approximate confidence intervals for the coefficient of variation.The confidence intervals for  in normal and lognormal were proposed by Koopmans et al. [24] and Verrill [25].Buntao and Niwitpong [26] also introduced an interval estimating the difference of the coefficient of variation for lognormal and delta-lognormal distributions.Curto and Pinto [27] constructed the confidence interval for  when random variables are not independently and identically distributed.Recent work of Gulhar et al. [28] has compared several confidence intervals for estimating the population coefficient of variation based on parametric, nonparametric, and modified methods.
However, the population mean may be known in several phenomena.The confidence intervals of the aforementioned authors have not been used for estimating the population coefficient of variation for the normal distribution with a known population mean.Therefore, our main aim in this paper is to propose three confidence intervals for  in a normal distribution with a known population mean.
The organization of this paper is as follows.In Section 2, the theoretical background of the proposed confidence intervals is discussed.The investigations of the performance of the proposed confidence interval through a Monte Carlo simulation study are presented in Section 3. A comparison of the confidence intervals is also illustrated by using an empirical application in Section 4. Conclusions are provided in the final section.

Theoretical Results
In this section, the mean and variance of the estimator of the coefficient of variation in a normal distribution with a known population mean are considered.In addition, we will introduce an unbiased estimator for the coefficient of variation, obtain its variance, and finally construct three confidence intervals: normal approximation, shortest-length, and equal-tailed confidence intervals.
Note that  +1 → 1 as  → ∞.Therefore, it follows that lim It means that κ0 is asymptotically unbiased and asymptotically consistent for  0 .From (10), the unbiased estimator of Using Lemma 1, the mean and variance of κ0 are given by Journal of Probability and Statistics Hence, κ0 is also asymptotically consistent for  0 .Next, we examine the accuracy of κ0 from another point view.Let us first consider the following theorem.

Shortest-Length Confidence
Converting the statement we can write Thus, the 100(1 − )% confidence interval for  0 based on the pivotal quantity  is where ,  > 0,  < , and the length of confidence interval for  0 is defined as In order to find the shortest-length confidence interval for  0 , the following problem has to be solved: where   is the probability density function of central chisquare distribution with  degrees of freedom.From Casella Table 1 is constructed for the numerical solutions of these equations by using the R statistical software [34][35][36].

Simulation Study
A Monte Carlo simulation was conducted using the R statistical software [34][35][36] version 3.0.1 to investigate the estimated coverage probabilities and expected lengths of three proposed confidence intervals and to compare them to the existing confidence intervals.The estimated coverage probability and the expected length (based on  replicates) are given by where #( ≤  ≤ ) denotes the number of simulation runs for which the population coefficient of variation  lies within the confidence interval.The data were generated from a normal distribution with a known population mean  0 = 10 and  0 = 0.05, 0.10, 0.20, 0.33, 0.50, and 0.67 and sample sizes () of 5, 10, 15, 25, 50, and 100.The number of simulation runs () is equal to 50,000 and the nominal confidence levels 1 −  are fixed at 0.90 and 0.95.Three existing confidence intervals are considered, namely, Miller's [7], McKay's [12], and Vangel's [18]. Miller: 0 ∈ (κ 0 −  ) .
The upper McKay's limit will have to be set to ∞ under the following condition [25]: and the upper Vangel's limit will have to be set to ∞ under the following condition: As can be seen from Tables 2 and 3, the three proposed confidence intervals have estimated coverage probabilities close to the nominal confidence level in all cases.On the other hand, the Miller's, McKay's, and Vangel's confidence intervals provide estimated coverage probabilities much different from the nominal confidence level, especially when the population coefficient of variation  0 is large.In other words, the estimated coverage probabilities of existing confidence intervals tend to be too high.Additionally, the estimated coverage probabilities of existing confidence intervals increase as the values of  0 get larger (i.e., for 95% McKay's confidence interval,  = 10, 0.9522 for  0 = 0.05; 0.9539 for  0 = 0.10; 0.9856 for  0 = 0.67).However, Figure 1 shows that the estimated coverage probabilities of the three proposed confidence intervals do not increase or decrease according to the values of  0 .
As can be seen from Figure 2, McKay's and Vangel's confidence intervals have longer expected lengths than Miller's

Figure 2 :
Figure 2: The expected lengths of 90% confidence intervals for the coefficient of variation in a normal distribution with a known population mean.

Table 1 :
The values of  and  for the shortest-length confidence interval for  0 .

Table 2 :
The estimated coverage probabilities and expected lengths of 90% confidence intervals for the coefficient of variation in a normal distribution with a known population mean.

Table 3 :
The estimated coverage probabilities and expected lengths of 95% confidence intervals for the coefficient of variation in a normal distribution with a known population mean.

Table 4 :
The 95% confidence intervals for the coefficient of variation of the weight of one-month old infants. 0 based on the pivotal quantity  is determined by the value of  and  satisfying