Extended Truncated Tweedie-Poisson Model

Abstract

Abstract. It has been argued that by truncating the sample space of the negative binomial and of the inverse Gaussian-Poisson mixture models at zero, one is allowed to extend the parameter space of the model. Here that is proved to be the case for the more general three parameter Tweedie-Poisson mixture model. It is also proved that the distributions in the extended part of the parameter space are not the zero truncation of mixed Poisson distributions and that, other than for the negative binomial, they are not mixtures of zero truncated Poisson distributions either. By extending the parameter space one can improve the fit when the frequency of one is larger and the right tail is heavier than is allowed by the unextended model. Considering the extended model also allows one to use the basic maximum likelihood based inference tools when parameter estimates fall in the extended part of the parameter space, and hence when the m.l.e. does not exist under the unextended model. This extended truncated Tweedie-Poisson model is proved to be useful in the analysis of words and species frequency count data.

Over‐and Underdispersion Models

“…Note that the excess of zeros (e.g., [36]) is a special case of overdispersion but then it is a very particular kind, measured by the zero-inflation index (see, e.g., [88][89][90]) and admitting various treatments (see, e.g., [51,85]). Zero-truncation and general truncation also produce the phenomenon of over-underdispersion (see, e.g., [105]), and we omit them here.…”

On zero-truncating and mixing Poisson distributions

“…In the following example we have a set of PGFs that correspond to the extended truncated negative binomial model of [5]. In part of the parameter space of that model it is both a ZTMP and an MZTP distribution, as described in [20], while in the rest of the parameter space, is an MZTP but not a ZTMP distribution. Example 1.…”

On zero-truncating and mixing Poisson distributions

“…In the following example we have a set of PGFs that correspond to the extended truncated negative binomial model of [5]. In part of the parameter space of that model it is both a ZTMP and an MZTP distribution, as described in [20], while in the rest of the parameter space, is an MZTP but not a ZTMP distribution. Example 1.…”

“…In Section 5 we compare these two sets of models based on the characterisations obtained, and in Section 6 we present some consequences of these characterisations. For an application of these characterisations to help describe a natural extension of the zero truncated Tweedie-Poisson mixture model, see [20].…”