Performance of Some Confidence Intervals for Estimating the Population Coefficient of Variation under both Symmetric and Skewed Distributions
Statistics, Optimization and Information Computing
This paper aims to compare the performance of proposed confidence intervals for population coefficient of variation (CV) with the existing confidence intervals, namely, McKay, Miller, and Gulher et al. confidence intervals under both symmetric and skewed distributions. We observed that the proposed augmented-large-sample (AA&K-ALS) confidence interval performed well in terms of coverage probability in all cases. The large-sample (A&A-LS) and adjusted degrees of freedom (AA&K-ADJ) confidence
... ADJ) confidence intervals had much lower coverage probability than the nominal level for skewed distributions. However, the average widths of the AA&K-LS confidence interval are narrower than that of the rest confidence intervals. Two real-life data are analyzed to illustrate the implementation of the several methods. 278 PERFORMANCE OF SOME CONFIDENCE INTERVALS the population parameter will be within this interval with a certain level of confidence as estimates for population parameters, while the hypothesis testing focuses on the use of statistical tests to accept or reject hypotheses concerning these parameters. The typical sample estimate of the population coefficient of variation (CV) is given as: where S is the sample standard deviation, the square root of the unbiased estimator of the population variance, andX is the sample mean. The point estimator of the population CV in (1) is a useful statistical measure, its confidence interval is more useful than the point estimator. In this paper, we choose the CV as a parameter of our interest because of its widespread use in describing the variation within a data set. Moreover, among scale parameters, the CV is a more informative quantity than others. As noted in , the CV is preferred to the variance or standard deviation in various fields of interest, especially in biological and medical research. The confidence interval for the CV given in literature is developed mostly based on the normality assumption. When the data are normally distributed, the coverage probability (CP) of this confidence interval is close to a nominal value of 1 − α. However, the underlying distributions are non-normal in many situations, like for example, the positively skewed data are common in real life, especially when sample sizes are small [1, 2, 3, 28] . In these situations, the CP of the confidence interval can be considerably below1 − α. Hummel  presented a confidence interval for the population variance by adjusting the degrees of freedom of the chi-square distribution. In order to develop approximate confidence intervals for variance under non-normality, Burch  considered a number of kurtosis estimators combined with large-sample. There are various methods available for estimating the confidence interval for a population CV. For more information on the confidence interval for CV, we refer to [15, 18, 24, 17, 27, 16, 7, 20, 25, 10, 21] and recently  among others. The necessary sample size for estimating a population parameter is important. Therefore, determining the sample size to estimate the population CV is also important. Tables of necessary sample sizes to have sufficiently narrow confidence intervals under different scenarios are provided by Kelley . The objective of this paper is to propose some new confidence intervals for estimating the population CV and compared them with some existing confidence intervals under the condition of symmetric and skewed distributions. A Monte-Carlo simulation will be conducted to compare the performance of the confidence intervals.