Asymptotic normality of the test statistics for the unified relative dispersion and relative variation indexes

Dispersion indexes with respect to the Poisson and binomial distributions are widely used to assess the conformity of the underlying distribution from an observed sample of the count with one or the other of these theoretical distributions. Recently, the exponential variation index has been proposed as an extension to nonnegative continuous data. This paper aims to gather to study the unified definition of these indexes with respect to the relative variability of a nonnegative natural exponential family of distributions through its variance function. We establish the strong consistency of the plug-in estimators of the indexes as well as their asymptotic normalities. Since the exact distributions of the estimators are not available in closed form, we consider the test of hypothesis relying on these estimators as test statistics with their asymptotic distributions. Simulation studies globally suggest good behaviours of these tests of hypothesis procedures. Applicable examples are analysed, including the lesser-known references such as negative binomial and inverse Gaussian, and improving the very usual case of the Poisson dispersion index. Concluding remarks are made with suggestions of possible extensions.