Discrete dispersion models and their Tweedie asymptotics

Abstract

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.

Appendices

Appendix A: Exponential dispersion models

In this appendix, we summarize some relevant facts about exponential dispersion models and Tweedie models. An exponential dispersion model $\mathrm {ED}(\mu ,\gamma )$ with mean $\mu \in \Omega$, dispersion parameter $\gamma >0$ and unit variance function $V(\mu )$ has PDF of the form

$$\begin{aligned} f(y;\mu ,\gamma )=a(y;\gamma )\exp \left[ -\frac{1}{2\gamma }\mathrm{d}(y;\mu )\right] \quad \text { for }\,y,\mu \in \Omega , \end{aligned}$$

(6.1)

where the unit deviance function $\mathrm{d}(y;\mu )$ is defined by

$$\begin{aligned} d(y;\mu )=2\int _{\mu }^{y}\frac{y-z}{V(z)}\,\mathrm{d}z\quad \text { for }\,y,\mu \in \Omega . \end{aligned}$$

The model (6.1) is, for each known value of $\gamma$, a natural exponential family with variance function $\gamma V(\mu )$. Hence, the function $a(y;\gamma )$ may be determined by Fourier inversion from the CGF, which may in turn be obtained from $V$. The model $\mathrm {ED}(\mu ,\gamma )$ satisfies the following reproductive property:

$$\begin{aligned} \overline{Y}_{n}\sim \mathrm {ED}(\mu ,\gamma /n), \end{aligned}$$

(6.2)

where $\overline{Y}_{n}$ is the average of $Y_{1},\ldots ,Y_{n}$, which are i.i.d. from $\mathrm {ED}(\mu ,\gamma )$.

The Tweedie exponential dispersion model $\mathrm {Tw}_{p}(\mu ,\gamma )$ has mean $\mu$ and unit variance function

$$\begin{aligned} V(\mu )=\mu ^{p}\quad \text {for }\, \mu \in \Omega _{p},\, \text {where }\, p\notin \left( 0,1\right) . \end{aligned}$$

The domain for $\mu$ is either $\Omega _{0}=R$ or $\Omega _{p}=R_{+}$ for $p\ne 0$. Tweedie models satisfy the scaling property

$$\begin{aligned} c\mathrm {Tw}_{p}(\mu ,\gamma )=\mathtt {\mathrm {Tw}}_{p}(c\mu ,c^{2-p}\gamma )\quad \text {for }\,c>0. \end{aligned}$$

(6.3)

Conventional Tweedie asymptotics (Jørgensen et al. 1994) have the following form. If $\mathtt {\mathrm {ED}}(\mu ,\gamma )$ with unit variance function $V(\mu )$ satisfies

$$\begin{aligned} V(\mu )\sim \mu ^{p} \quad \text { as }\,\mu \downarrow 0\text { or }\mu \rightarrow \infty \end{aligned}$$

then

$$\begin{aligned} c^{-1}\mathtt {\mathrm {ED}}(c\mu ,c^{2-p}\gamma )\overset{D}{\rightarrow } \mathrm {Tw}_{p}(\mu ,\gamma ) \quad \text {as }\,c\downarrow 0 \text { or }c\rightarrow \infty \, , \end{aligned}$$

(6.4)

respectively. The proof is based on convergence of the variance function on the left-hand side of (6.4),

$$\begin{aligned} c^{-2}c^{2-p}\gamma V(c\mu )\rightarrow \gamma \mu ^{p}, \end{aligned}$$

applying Mora’s (1990) convergence theorem. The case $c^{2-p}\rightarrow \infty$ requires the model $\mathtt {\mathrm {ED}}(\mu ,\gamma )$ to be infinite divisible. This result implies a Tweedie approximation, by means of (6.3)

$$\begin{aligned} \mathtt {\mathrm {ED}}(c\mu ,c^{2-p}\gamma )\overset{\cdot }{\sim }\mathtt { \mathrm {Tw}}_{p}(c\mu ,c^{2-p}\gamma )\,\quad \text {as }\,c\downarrow 0 \text { or }c\rightarrow \infty . \end{aligned}$$

In some cases, we have a large-sample interpretation of Tweedie convergence. Let us consider the average $\bar{Y}_{n}$ with distribution (6.2). Then for $p\ne 2$ we obtain

$$\begin{aligned} n^{-1/(p-2)}\mathtt {\mathrm {ED}}(n^{1/(p-2)}\mu ,\gamma /n)\overset{D}{ \rightarrow } \mathrm {Tw}_{p}(\mu ,\gamma )\quad \text {as }\, n\rightarrow \infty . \end{aligned}$$

We interpret this result as saying that the scaled and exponentially tilted average $\bar{Y}_{n}$ converges to a Tweedie distribution as $n\rightarrow \infty$.

Appendix B: Proof of Theorem 4.1

Consider a sequence of factorial tilting families $\mathrm {FT}_{n}(\mu )$ with local dispersion functions $v_{n}$ having domain $\Psi _{n}$ and FCGF $C_{n}$ satisfying the conditions of Theorem 4.1. The idea of the proof is to obtain the FCGF derivative $\dot{C}$ from the limiting dispersion function $v$, and in turn use the uniform convergence to show convergence of the sequence $C_{n}$.

We begin by considering the nonzero case, where $v(\mu )\ne 0$ for $\mu \in \Psi _{0}$. Let $K$ be a given compact subinterval of $\Psi _{0}$. By assumption $\Psi _{0}\subseteq \mathrm {int}\left( \lim \Psi _{n}\right)$, so we may assume that $K\subseteq \Psi _{n}$ from some $n_{0}$ on. We only need to consider $n>n_{0}$ from now on. Fix a $\mu _{0}\in \mathrm {int}\,K$. Let $\psi _{n}=\dot{C}_{n}^{-1}$ denote the inverse FCGF derivative defined by $\dot{\psi }_{n}\left( \mu \right) =1/v_{n}(\mu )$ on $\Psi _{n}$ and $\psi _{n}\left( \mu _{0}\right) =0$. Let $\dot{C}_{n},\, C_{n}$, etc., denote the quantities associated with this parametrization. Similarly, define $\psi :\Psi _{0}\rightarrow \mathbb {R}$ by $\dot{\psi }\left( \mu \right) =1/v(\mu )$ on $\Psi _{0}$ and $\psi (\mu _{0})=0$. Then for $\mu \in K$

$$\begin{aligned} \left| \dot{\psi }_{n}\left( \mu \right) -\dot{\psi }\left( \mu \right) \right| =\frac{\left| v_{n}(\mu )-v(\mu )\right| }{v_{n}(\mu )v(\mu )}. \end{aligned}$$

(6.5)

By the uniform convergence of $v_{n}(\mu )$ to $v(\mu )$ on $K$, it follows that $\left\{ v_{n}(\mu )\right\}$ is uniformly bounded on $K$. Since $v(\mu )$ is bounded on $K$, it follows from (6.5) and from the uniform convergence of $v_{n}$ that $\dot{\psi }_{n}\left( \mu \right) \rightarrow \dot{\psi }\left( \mu \right)$ uniformly on $K$. This and the fact that $\psi _{n}\left( \mu _{0}\right) =\psi (\mu _{0})$ for all $n$ implies, by a result from Rudin (1976, Theorem 7.17), that $\psi _{n}\left( \mu \right) \rightarrow \psi \left( \mu \right)$ uniformly on $K$. Since $K$ was arbitrary, we have $\psi _{n}\left( \mu \right) \rightarrow \psi \left( \mu \right)$ for all $\mu \in \Psi _{0}$.

Let $I_{n}=\psi _{n}\left( \Psi _{n}\right)$ and $I_{0}=\psi (\Psi _{0})\subseteq \mathrm {int}\left( \lim I_{n}\right)$. Let $J=\psi (K)\subseteq I_{0}$ and $J_{n}=\psi _{n}(K)\subseteq I_{n}$. Define $\dot{C} :I_{0}\rightarrow \Psi _{0}$ by $\dot{C}(y)=\psi ^{-1}(y)$. Since $\psi$ is strictly monotone and differentiable, the same is the case for $\dot{C}$. Let $\mu \in K$ be given and let $y=\psi (\mu )\in J$ and $y_{n}=\psi _{n}(\mu )\in J_{n}$. Since $v_{n}(\mu )$ is uniformly bounded on $K$, there exists an $M>0$ such that $\left| v_{n}(\mu )\right| \le M$ for all $n$ and $\mu \in K$. It follows that $\left| \ddot{C}_{n}(y)\right| =\left| v_{n}\left( \dot{C}_{n}(y)\right) \right| \le M$ for all $y\in J$ due to the fact that $J\subseteq J_{n}$ for $n$ large enough. Since $\mu =\dot{C}(y)=\dot{C}_{n}(y_{n})$ we find, using the mean value theorem, that

$$\begin{aligned} \left| \dot{C}_{n}(y)-\dot{C}(y)\right|= & {} \left| \dot{C}_{n}(y)- \dot{C}_{n}(y_{n})\right| \\\le & {} M\left| y-y_{n}\right| \\= & {} M\left| \psi (\mu )-\psi _{n}(\mu )\right| . \end{aligned}$$

This implies that $\dot{C}_{n}(y)\rightarrow \dot{C}(y)$ uniformly in $y\in J$. Since $C(0)=C_{n}(0)$ for all $n$, it follows by similar arguments as above that $C_{n}(y)\rightarrow C(y)$ uniformly on $J$. We conclude from the convergence of the sequence of MGFs $\exp \left[ C_{n}\left( e^{s}-1\right) \right] \rightarrow \exp \left[ C\left( e^{s}-1\right) \right]$ for $s\in \log \left( J+1\right)$ that the sequence of distributions $\mathrm {FT}_{n}(\mu _{0})$ converges weakly to a probability measure $P$ with FCGF $C$. We let $\mathrm {FT}(\mu )$ denote the factorial tilting family generated by $P$ with local dispersion function $v$ on $\Psi _{0}$. We may now complete the proof in the nonzero case by proceeding like in the proof of Proposition 2.1.

In the case where $v(\mu )=0$ (the zero case), we cannot define the function $\psi$ as above. Instead we take $C(t)=t\mu _{0},$ such that $\dot{C} (t)=\mu _{0}$ and $\ddot{C}(t)=0$ for $t\in \mathbb {R}$. For any $\epsilon >0$, we may choose an $n_{0}$ such that $\left| v_{n}(\mu )\right| \le \epsilon$ for any $n\ge n_{0}$ and $\mu \in K$. For such $n$ and $\mu$, we hence obtain

$$\begin{aligned} \left| \psi _{n}(\mu )\right| =\int _{\mu _{0}}^{\mu }\frac{1}{ \left| v_{n}(t)\right| }\,\mathrm{d}t\ge \frac{\left| \mu -\mu _{0}\right| }{\epsilon }, \end{aligned}$$

which can be made arbitrarily large by choosing $\epsilon$ small. We hence conclude that $J_{n}=\psi _{n}(K)\rightarrow \mathbb {R}$ as $n\rightarrow \infty$.

Now we let $J$ be a compact interval such that $0\in \mathrm {int}J$, implying that $J\subseteq \,J_{n}$ for $n$ large enough. For such $n$ we hence obtain that $\left| \ddot{C}_{n}(t)\right| =\left| v_{n}( \dot{C}_{n}(t))\right| \le \epsilon$ for all $t\in J$, because then $\dot{C}_{n}(t)\in K$. Since $\mu _{0}=\dot{C}(t)=\dot{C}_{n}(0)$ we find, again by the mean value theorem, that for $t\in J$,

$$\begin{aligned} \left| \dot{C}_{n}(t)-\dot{C}(t)\right| =\left| \dot{C}_{n}(t)- \dot{C}_{n}(0)\right| \le \epsilon \left| t\right| . \end{aligned}$$

This implies that $\dot{C}_{n}(t)\rightarrow \dot{C}(t)$ uniformly in $t\in J$. By similar arguments as above, we conclude that $\mathrm {FT}_{n}(\mu _{0})$ converges weakly to a probability measure $P$ with FCGF $C(t)=t\mu _{0}$, which implies the desired conclusion in the zero case, completing the proof.