Discrete dispersion models and their Tweedie asymptotics

By Bent Jørgensen, Célestin C. Kokonendji. 2014

73 citations recorded by the source corpus.

We introduce a class of two-parameter discrete dispersion models, obtained by combining convolution with a factorial tilting operation, similar to exponential dispersion models which combine convolution and exponential tilting. The equidispersed Poisson model has a special place in this approach, whereas several overdispersed discrete distributions, such as the Neyman Type A, Pólya–Aeppli, negative binomial and Poisson-inverse Gaussian, turn out to be Poisson–Tweedie factorial dispersion models with power dispersion functions, analogous to ordinary Tweedie exponential dispersion models with power variance functions. Using the factorial cumulant generating function as tool, we introduce a dilation operation as a discrete analogue of scaling, generalizing binomial thinning. The Poisson–Tweedie factorial dispersion models are closed under dilation, which in turn leads to a Poisson–Tweedie asymptotic framework where Poisson–Tweedie models appear as dilation limits. This unifies many discrete convergence results and leads to Poisson and Hermite convergence results, similar to the law of large numbers and the central limit theorem, respectively. The dilation operator also leads to a duality transformation which in some cases transforms overdispersion into underdispersion and vice versa. Finally, we consider the multivariate factorial cumulant generating function, and introduce a multivariate notion of over- and underdispersion, and a multivariate zero inflation index.

Subfields: Artificial Intelligence, Economics and Econometrics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f29b67f70a0ea47d268c36ef244bc82b40161e30

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Wagner Hugo Bonat
2. Bent Jørgensen
3. John P. Hinde
4. Clarice G. B. Dem'etrio
5. Célestin C. Kokonendji

In Economics and Econometrics

1. Wagner Hugo Bonat
2. Célestin C. Kokonendji
3. Bent Jørgensen
4. John P. Hinde
5. Clarice G. B. Dem'etrio
6. Mario V. Wüthrich

In Statistics and Probability

1. Wagner Hugo Bonat
2. Célestin C. Kokonendji
3. John P. Hinde
4. Clarice G. B. Dem'etrio
5. Bent Jørgensen
6. Dan Hall

Citations in the corpus

• Multivariate covariance generalized linear models (2015, background): “The Poisson-Tweedie family has variance function ϑ(µ; p) = µ + µp, and many important models for count data are special cases, for example the Hermite (p = 0), Neyman Type A (p = 1), negative binomial (p = 2) and Poissoninverse Gaussian (p = 3), see Jørgensen and Kokonendji (2014).”
• Extended Poisson–Tweedie: Properties and regression models for count data (2016, methodology): “Jørgensen and Kokonendji (2016) showed by using factorial cumulant function that for Y ∼ PTwp( , ), E(Y) = and Var(Y) = + p.”