The distribution of each member of the family of statistics based on the $R_{\phi }$-divergence for testing goodness-of-fit is a chi-squared to $o\left(1\right)$ (Pardo [pard96]). In this paper a closer approximation to the exact distribution is obtained by extracting the $\phi$-dependent second order component from the $o\left(1\right)$ term.

In this paper we investigate two classes of exponential dispersion models (EDMs) for overdispersed count data with respect to the Poisson distribution. The first is a class of Poisson mixture with positive Tweedie mixing distributions. As an approximation (in terms of unit variance function) of the first, the second is a new class of EDMs characterized by their unit variance functions of the form μ + μp, where p is a real index related to a precise model. These two classes provide some alternatives...