On interval estimation of the coefficient of variation for the three-parameter Weibull, lognormal and gamma distribution: A simulation-based approach

Abstract

The coefficient of variation (CV) of a population is defined as the ratio of the population standard deviation to the population mean. It is regarded as a measure of stability or uncertainty, and can indicate the relative dispersion of data in the population to the population mean. CV is a dimensionless measure of scatter or dispersion and is readily interpretable, as opposed to other commonly used measures such as standard deviation, mean absolute deviation or error factor, which are only interpretable for the lognormal distribution. CV is often estimated by the ratio of the sample standard deviation to the sample mean, called the sample CV. Even for the normal distribution, the exact distribution of the sample CV is difficult to obtain, and hence it is difficult to draw inferences regarding the population CV in the frequentist frame. Different methods of estimating the sample standard deviation as well as the sample mean result in different shapes of the sampling distribution of the sample CV, from which inferences about the population CV can be made. In this paper we propose a simulation-based Bayesian approach to tackle this problem. A set of real data is used to generate the sampling distribution of the CV under the assumption that the data follow the three-parameter Gamma distribution. A probability interval is then constructed. The method also applies easily to lognormal and Weibull distributions.