Approximate Confidence Limits for the Coefficient of Variation of Gamma Distributions

In this paper a procedure is described for finding approximate confidence limits for the coefficient of variation if a sample of size n from a gamma distribution (formula (1), below) is given. In many instances the variables with which one is dealing are nonnegative. In some of these cases the normal distribution cannot be assumed, either because it obviously fits badly, or else because admitting the possibility of negative values is embarrassing. In such cases the obvious next choice is either the lognormnal distribution or the gamma distribution (sometimes also called the Pearson type III distribution, or the distribution of ax'). Both distributions are similar in form and have two parameters. The advantage of the lognormal distribution is that normal theory can be applied once the variables have been transformed. However, it can happen that one does not wish to analyse transformned variables, and in such cases the gamma distribution is preferable, because it is easier to handle analytically. In many fields of application the coefficient of variation is more popular as a descriptive parameter than the variance or standard deviation as such. This is so because often the coefficient of variatioll, but not the variance, remains invariant if one shifts to a distribution with another mean. This is another reason for using a gamma distribution; in fact, one of its parameters determines the coefficient of variation, whereas the variance depends on both parameters.