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Dispersion Models

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A dispersion model, denoted $Y\sim \mathrm{DM}(\mu ,\sigma ^{2}),$ is a two-parameter family of distributions with probability density functions on $\mathbb{R}$ of the form

$\displaystyle f(y;\mu ,\sigma ^{2})=a(y;\sigma ^{2})\exp \left\{ -\frac{1}{2\sigma ^{2}}<tex2html_comment_mark>12 d(y;\mu )\right\} .$

(1)

Here $\mu$ and $\sigma ^{2}$ are real parameters with domain $(\mu ,\sigma ^{2})\in \Omega \times \mathbb{R}_{+}$ ( $\Omega$ being an interval), called the positionand dispersionparameters, respectively

Also

and

are suitable functions such that (1) is a probability density function for all parameter values. In particular,

is assumed to be a unit deviance, satisfying $d(\mu ;\mu )=0$ for $\mu \in \Omega$ and $d(y;\mu )>0$ for $y\neq \mu .$ Dispersion models were introduced by Sweeting [23] and Jørgensen [7], [10], who extended the analysis of deviance for generalized linear models in the sense of Nelder and Wedderburn [21] to non-linear regression models with error distribution $\mathrm{DM}(\mu ,\sigma ^{2})$ .

In many cases, the unit deviance is regular, meaning that it is twice continuously differentiable and $\partial ^{2}d/\partial \mu ^{2}(\mu ;\mu )>0$ for $\mu \in \Omega$ . We then define the unit variance function for $\mu \in \Omega$ by

$\displaystyle V(\mu )=\frac{2}{\frac{\partial ^{2}d}{\partial \mu ^{2}}(\mu ;\mu )}$ .

(2)

For example, the normal distribution $\mathrm{N}(\mu ,\sigma ^{2})$ corresponds to the unit deviance $d(y;\mu )=\left( y-\mu \right) ^{2}$ on $% \mathbb{R}^{2}$ , with unit variance function $V(\mu )\equiv 1$ and $% a(y;\sigma ^{2})=\left( 2\pi \sigma ^{2}\right) ^{-1/2}.$ Note that for $% 0<\rho <2$ the unit deviance $d(y;\mu )=\vert y-\mu \vert^{\rho }$ , with $a(y;\sigma ^{2})$ constant (not depending on

) provides an example of a dispersion model where

is not regular.

The renormalized saddlepoint approximation for a dispersion model with regular unit deviance is defined for $y\in \Omega$ by

$\displaystyle f(y;\mu ,\sigma ^{2})\sim a_{0}(\mu ,\sigma ^{2})V^{-\frac{1}{2}}(y)\exp \left\{ -\frac{1}{2\sigma ^{2}}d(y;\mu )\right\} ,$

(3)

where $a_{0}(\mu ,\sigma ^{2})$ is a normalizing constant, making the right-hand side of (3) a probability density function on $\Omega$ with respect to Lebesgue measure. The ordinary saddlepoint approximation is obtained by the asymptotic approximation $a_{0}(\mu ,\sigma ^{2})\sim \left( 2\pi \sigma ^{2}\right) ^{-1/2}$ as $\sigma ^{2}\downarrow 0$ , and correspondingly, the distribution $\mathrm{DM}(\mu ,\sigma ^{2})$ is asymptotically normal $\mathrm{N}\left\{ \mu ,\sigma ^{2}V(\mu )\right\}$ for $\sigma ^{2}$ small, so that $\mu$ and $\sigma ^{2}V(\mu )$ are the asymptotic mean and variance of

, respectively. In this way, dispersion models have properties that resemble those of the normal distribution. In particular the unit deviance $d(y;\mu )$ is often a measure of squared distance between

and $\mu$ .

There are two main classes of dispersion models, namely proper dispersion models ( $\mathrm{PD}$ ) and reproductive exponential dispersion models ( $\mathrm{ED}$ ), as defined below. The class of additive exponential dispersion models ( $\mathrm{ED}^{\ast }$ ) is not of the dispersion model form (1), but is closely related to reproductive exponential dispersion models. These three types of models cover a comprehensive range of non-normal distributions, and as shown in Table 1 they include many standard statistical families as special cases. The exponential and Poisson families are examples of natural exponential families, which correspond to exponential dispersion models with $\sigma ^{2}=1$ .

**Table 1:** Main examples of univariate dispersion models.
Data type	Support	Examples	Type
Real data	$\mathbb{R}$	Normal, generalized hyperbolic secant	$\mathrm{% ED}$
Positive data	$\mathbb{R}_{+}$	Exponential, gamma, inverse Gaussian	$% \mathrm{ED}$
Positive data with zeros	$\mathbb{R}_{0}=[0,\infty )$	Compound Poisson distribution	$\mathrm{ED}$
Proportions		Simplex, Leipnik distributions	$\mathrm{PD}$
Directional data	$[0,2\pi )$	von Mises distribution	$\mathrm{PD}$
Count data	$\mathbb{N}_{0}=\{0,1,2,\dots \}$	Poisson, negative binomial distributions	$\mathrm{ED}^{\ast }$
Binomial count data	$\{0,1,\dots ,m\}$	Binomial distribution	$\mathrm{% ED}^{\ast }$

Proper dispersion models

Jørgensen [12] proposed a special case of (1), called a proper dispersion model, which corresponds to a density of the form

$\displaystyle f(y;\mu ,\sigma ^{2})=a(\sigma ^{2})V^{-\frac{1}{2}}(y)\exp \left\{ -\frac{1}{2\sigma ^{2}}d(y;\mu )\right\} ,$

(4)

where

is a given regular unit deviance with unit variance function $% V(\mu )$ . We denote this model by $Y\sim \mathrm{PD}(\mu ,\sigma ^{2})$ . Note that the unit deviance

characterizes the proper dispersion model (4), because the unit variance function (2) is a function of

, while $a(\sigma ^{2})$ , in turn, is a normalizing constant. It also follows that the normalizing constant of (3) is $% a_{0}(\mu ,\sigma ^{2})=a(\sigma ^{2})$ , making the the two sides of (3) identical in this case. Conditions under which a given unit deviance gives rise to a proper dispersion model, and the connection with exactness of Barndorff-Nielsen's $p^{\ast }$ -formula, are discussed by Jørgensen [13, Ch. 5].

Many proper dispersion models are transformation models when $\sigma ^{2}$ is known. For example, unit deviances of the form $d(y;\mu )=h(y-\mu )$ on $% \Omega =\mathbb{R}$ , for a suitable function give rise to so-called location-dispersion models, where the unit variance function is constant. A specific example is given by the unit deviance $d(y;\mu )=\log \left\{ 1+\left( y-\mu \right) ^{2}\right\} ,$ which corresponds to the Student location family. Another important case corresponds to unit deviances of the form $d(y;\mu )=h(y/\mu )$ on $\Omega =\mathbb{R}_{+}$ , which give rise to so-called scale-dispersion models. For example, Jørgensen [13, Ch. 5] showed that the generalized inverse Gaussian distribution is a family of scale-dispersion models with unit deviances

$\displaystyle d_{\beta }(y;\mu )=2\beta \log \frac{\mu }{y}+\left( 1+\beta \right) \frac{y}{\mu }+\left( 1-\beta \right) \frac{\mu }{y}-2,$

(5)

for $y,\mu >0$ , where $\beta \in \lbrack -1,1]$ is an additional parameter. In particular, the values $\beta =\pm 1$ correspond to the gamma and reciprocal gamma families, respectively, and $\beta =0$ to the one-dimensional hyperboloid distribution. The von Mises distribution, corresponding to the unit deviance $d(y;\mu )=2\left\{ 1-\cos \left( y-\mu \right) \right\}$ on $\Omega =\left( 0,2\pi \right)$ is also a transformation model when $\sigma ^{2}$ is known, corresponding to the group of additions modulo $2\pi$ .

The class of simplex distributions of Barndorff-Nielsen and Jørgensen [2] contains several proper dispersion models as special cases. The simplest example is given by the probability density function

$\displaystyle f(y;\mu ,\sigma ^{2})=\frac{1}{\sigma \sqrt{2\pi }}\{y(1-y)\}^{-3... ... \left\{ -\frac{(y-\mu )^{2}}{2\sigma ^{2}y(1-y)\mu ^{2}(1-\mu )^{2}}\right\} ,$

(6)

where $y,\mu \in (0,1)$ and $\ \sigma ^{2}>0$ . The transformed Leipnik distribution, defined by the unit deviance

$\displaystyle d(y;\mu )=\log \left\{ 1+\frac{(y-\mu )^{2}}{y(1-y)}\right\}$ for $\displaystyle <tex2html_comment_mark>41 y,\mu \in (0,1)$

provides a further examples of proper dispersion models. This and the simplex distribution are examples of proper dispersion models that are not transformation models when $\sigma ^{2}$ is known.

Exponential dispersion models

Exponential dispersion models are two-parameter extensions of natural exponential families. A natural exponential family has densities of the form

$\displaystyle f(x;\theta )=c(x)\exp \left\{ \theta x-\kappa (\theta )\right\}$ ,

(7)

with respect to a suitable measure $\nu$ , usually Lebesgue or counting measure. Here

is a given function and $\kappa (\theta )=\log \int c(x)e^{\theta x}\,\nu (dx)$ is the corresponding cumulant function.The canonical parameter $\theta$ has domain $\Theta$ ; the largest interval for which $\kappa (\theta )$ is finite. The mean of a random variable

with distribution (7) is $\mu =\tau (\theta )$ , where $\tau (\theta )=\kappa ^{\prime }(\theta )$ (a one-to-one function of $% \theta$ ), and the domain for $\mu$ is $\Omega =\tau (\mathrm{int}\Theta )$ , where $\mathrm{int}\Theta$ denotes the interior of the domain $\Theta$ . If necessary, we define the value of $\mu$ by continuity at boundary points of $\Theta$ . Examples of natural exponential families include the exponential, geometric, Poisson and logarithmic families.

The variance of is $V(\mu )=\tau ^{\prime }\left\{ \tau ^{-1}(\mu )\right\}$ , where is known as the variance function of the family (7). This function characterizes (7) among all natural exponential families, see for example Morris [20], and is an important convergence and characterization tool for natural exponential families, see e.g. Jørgensen [13, Ch. 2].

An additive exponential dispersion model is a two-parameter extension of natural exponential families with densities of the form

$\displaystyle f^{\ast }(x;\theta ,\lambda )=c^{\ast }(x;\lambda )\exp \left\{ \theta x-\lambda \kappa (\theta )\right\} ,$

(8)

where the index parameter $\lambda$ has domain $\Lambda$ consisting of those values of $\lambda >0$ for which $\lambda \kappa (\theta )$ is a cumulant function of some function $c^{\ast }(x;\lambda )$ . The mean and variance of a random variable

with distribution (8) are $\lambda \mu$ and $\lambda V(\mu )$ , respectively. Many discrete distributions are additive exponential dispersion models, for example the binomial and negative binomial families. An additive exponential dispersion model is denoted $X\sim \mathrm{ED}^{\ast }(\mu ,\lambda ).$

A reproductive exponential dispersion model is defined by applying the transformation $Y=X/\lambda$ to (8), known as the duality transformation. The reproductive form is denoted $Y\sim \mathrm{ED}(\mu ,\sigma ^{2})$ , corresponding to densities of the form

$\displaystyle f(y;\theta ,\lambda )=c(y;\lambda )\exp [\lambda \{\theta y-\kappa (\theta )\}]$

(9)

for a suitable function $c(y;\lambda ).$ Here $\mu$ (with domain $\Omega$ ) is the mean of

, $\sigma ^{2}=1/\lambda$ is the dispersion parameter, and the variance of

is $\sigma ^{2}V(\mu ).$ Reproductive exponential dispersion models were proposed by Tweedie [24] in 1947, and again in 1972 by Nelder and Wedderburn [21] as the class of error distributions for generalized linear models. The inverse duality transformation $% X=\lambda Y$ takes (9) back into the form (8) so that each exponential dispersion model has, in principle, an additive as well as a reproductive form. Both (9) and (8) are natural exponential families when the index parameter $\lambda$ is known.

To see the connection with dispersion models as defined by (1), we introduce the unit deviance corresponding to (9),

$\displaystyle d(y;\mu )=2\left[ \sup_{\theta }\left\{ \theta y-\kappa (\theta )\right\} -y\tau ^{-1}(\mu )+\kappa \left\{ \tau ^{-1}(\mu )\right\} \right] ,$

(10)

which is a Kullback-Leibler distance, see Hastie [4]. Defining

$\displaystyle a(y;\sigma ^{2})=c(y;\sigma ^{-2})\exp \left[ \sup_{\theta }\left\{ \theta y-\kappa (\theta )\right\} \right] ,$

we may write the density (9) in the dispersion model form (1). Consequently, reproductive exponential dispersion models form a sub-class of dispersion models, whereas additive exponential dispersion models are not in general of the form (9).

The overlap between exponential and proper dispersion models is small; in fact the normal, gamma and inverse Gaussian families are the only examples of exponential dispersion models that are also proper dispersion models.

An additive exponential dispersion model satisfies a convolution formula, defined as follows. If $X_{1},\dots ,X_{n}$ are independent random variables, and $X_{i}\sim \mathrm{ED}^{\ast }(\mu ,\lambda _{i})$ with $% \lambda _{i}\in \Lambda$ for $i=1,\dots ,n$ , then the distribution of $% X_{+}=X_{1}+\cdots +X_{n}$ is

$\displaystyle X_{+}\sim \mathrm{ED}^{\ast }(\mu ,\lambda _{1}+\cdots +\lambda _{n}),$

(11)

where in fact $\lambda _{1}+\cdots +\lambda _{n}\in \Lambda .$ This is called the additive property of an additive exponential dispersion model.

An additive exponential dispersion model $\mathrm{ED}^{\ast }\left( \mu ,\lambda \right)$ gives rise to a stochastic process { $t\in \Lambda \cup \left\{ 0\right\}$ } with stationary and independent increments, called an additive process. This process is defined by assuming that , along with the following distribution of the increments:

$\displaystyle X(t+s)-X(t)\sim \mathrm{ED}^{\ast }\left( \mu ,s\right) ,$

for $s,t\in \Lambda$ . An additive process with time domain $\Lambda =% \mathbb{N}$ (the positive integers) is a random walk, the simplest example being the Bernoulli process. An additive process with time domain $\Lambda =% \mathbb{R}_{+}$ (which requires the distribution to be infinitely divisible), is a Lévy process, including examples such as Brownian motion with drift and the Poisson process. Note, in particular, that the average of the process over the interval

has distribution $% X(t)/t\sim \mathrm{ED}(\mu ,t^{-1})$ , which follows by an application of the duality transformation.

A reproductive exponential dispersion model $\mathrm{ED}(\mu ,\sigma ^{2})$ satisfies the following reproductive property, which follows as a corollary to (11). Assume that $Y_{1},\dots ,Y_{n}$ are independent and $Y_{i}\sim \mathrm{ED}\left( \mu ,\sigma ^{2}/w_{i}\right)$ for $% \,i=1,\dots ,n,$ where $w_{1},\ldots ,w_{n}$ are positive weights such that $% w_{i}/\sigma ^{2}\in \Lambda$ for all . Then, with $w_{+}=w_{1}+\cdots +w_{n}$ ,

$\displaystyle \frac{1}{w_{+}}\sum_{i=1}^{n}w_{i}Y_{i}\sim \mathrm{ED}\left( \mu ,\frac{<tex2html_comment_mark>61 \sigma ^{2}}{w_{+}}\right)$ .

(12)

Tweedie exponential dispersion models

The class of Tweedie models, denoted $\mathrm{Tw}_{p}(\mu ,\sigma ^{2}),$ consist of reproductive exponential dispersion models corresponding to the unit variance functions $V(\mu )=\mu ^{p},$ where is a parameter with domain $(-\infty ,0]\cup \lbrack 1,\infty ]$ and $\Omega$ is defined in Table 2. Here we let $p=\infty$ correspond to the variance function $V(\mu )=e^{\beta \mu }$ for some $\beta \in \mathbb{R}$ . These models were introduced independently by Tweedie [25], Morris [19], Hougaard [5] and Bar-Lev and Enis [1]. As shown in Table 2, the Tweedie class contains several well-known families of distributions. For or the Tweedie models are natural exponential families generated by extreme or positive stable distributions with index $\alpha =(p-2)/(p-1).$ For $p\neq 0,1,2,$ the unit deviance of the family $\mathrm{Tw}_{p}(\mu ,\sigma ^{2})$ is given by

$\displaystyle d_{p}(y;\mu )=2\left[ \frac{\left\{ \max (y,0)\right\} ^{2-p}}{(1... ...<tex2html_comment_mark>64 \frac{y\mu ^{1-p}}{1-p}+\frac{\mu ^{2-p}}{2-p}\right]$ ,

where

belongs to the support, see Table 2.

**Table 2:** Summary of Tweedie exponential dispersion models
Distribution family		Support	$\Omega$	$\;\;\;\Theta$
Extreme stable exponential family		$\mathbb{R}$	$\mathbb{R}_{+}$	$\;\;\;\mathbb{R}_{0}$
Normal distribution		$\mathbb{R}$	$\mathbb{R}$	$\;\;\;\mathbb{R}$
Poisson distribution		$\mathbb{N}_{0}$	$\mathbb{R}_{+}$	$\;\;\;% \mathbb{R}$
Compound Poisson distribution		$\mathbb{R}_{0}$	$\mathbb{R}_{+}$	$\;\;\;\mathbb{R}_{-}$
Gamma distribution		$\mathbb{R}_{+}$	$\mathbb{R}_{+}$	$\;\;\;% \mathbb{R}_{-}$
Positive stable exponential family		$\mathbb{R}_{+}$	$\mathbb{R}% _{+}$	$-\mathbb{R}_{0}$
Inverse Gaussian distribution		$\mathbb{R}_{+}$	$\mathbb{R}_{+}$	$-\mathbb{R}_{0}$
Extreme stable exponential family	$p=\infty$	$\mathbb{R}$	$\mathbb{R}$	$\;\;\;\mathbb{R}_{-}$
Notation: $-\mathbb{R}_{0}=(-\infty ,0]$

The Tweedie models are characterized by the following scaling property:

$\displaystyle b\ \mathrm{Tw}_{p}(\mu ,\sigma ^{2})\sim \mathrm{Tw}_{p}(b\mu ,b^{2-p}\sigma ^{2}).$

(13)

for all

see Jørgensen, Martínez and Tsao [15] and Jørgensen [13, Ch. 4]. The former authors showed that the scaling property implies a kind of central limit theorem for exponential dispersion models, where Tweedie models appear as limiting distributions.

For , the Tweedie models are compound Poisson distributions, which are continuous non-negative distributions with an atom at zero. Such models are useful for measurements of for example precipitation, where wet periods show positive amounts, while dry periods are recorded as zeros. Similarly, the total claim on an insurance policy over a fixed time interval (Renshaw [22], Jørgensen and Souza [17]), may be either positive if claims were made in the period or zero if no claims were made. Other values of (except ) correspond to continuous distributions with support either $\mathbb{R}_{+}$ or $\mathbb{R}$ .

Let us apply the inverse duality transformation $X=\lambda Y$ to the Tweedie variable $Y\sim \mathrm{Tw}_{p}(\mu ,1/\lambda )$ , in order to obtain the additive form of the Tweedie distribution. By the scaling property (13) this gives $X\sim \mathrm{Tw}_{p}(\lambda \mu ,\lambda ^{1-p})$ , which shows that the Tweedie exponential dispersion models have an additive as well as a reproductive form. The additive form of the Tweedie model gives rise to a class of additive processes $\left\{ X(t):t\geq 0\right\}$ , defined by the following distribution of the increments:

$\displaystyle X(t+s)-X(t)\sim \mathrm{Tw}_{p}\left( s\mu ,s^{1-p}\right) ,$

for

. This class of Hougaard Lévy processes was introduced by Jørgensen [11] and Lee and Whitmore [18]. Brownian motion with drift (

) and the Poisson process (

) mentioned above are examples of Hougaard processes, and other familiar examples include the gamma process (

) and the inverse Gaussian process.(

). The Hougaard processes corresponding to

are compound Poisson processes with gamma distributed jumps.

Multivariate dispersion models

One way to generalize (1) to the multivariate case is to consider a probability density function of the form

$\displaystyle f(\boldsymbol{y};\boldsymbol{\mu },\sigma ^{2})=a(\boldsymbol{y};... ...igma ^{2}}d(\boldsymbol{y};\boldsymbol{\mu }<tex2html_comment_mark>76 )\right\}$ for $\displaystyle \boldsymbol{y}\in \mathbb{R}^{k}$ ,

(14)

where $(\boldsymbol{\mu },\sigma ^{2})\in \Omega \times \mathbb{R}_{+}$ , and $\Omega$ is now a domain in $\mathbb{R}^{k}.$ Here

is a suitable function such that (14) is a probability density function, and

is again a unit deviance satisfying $d(\boldsymbol{\mu ;\mu })=0$ for $% \boldsymbol{\mu }\in \Omega$ and $d(\boldsymbol{y;\mu })>0$ for $% \boldsymbol{y}\neq \boldsymbol{\mu }$ . The multivariate exponential dispersion models of Jørgensen [8], [9] provide examples of (14). Other examples include the multivariate von Mises-Fisher distribution, the multivariate hyperboloid distribution (see Jensen [6]), and some special cases of the multivariate simplex distributions of Barndorff-Nielsen and Jørgensen [2].

A more flexible definition of multivariate dispersion models is obtained by considering models of the form

$\displaystyle f(\boldsymbol{y};\boldsymbol{\mu },\boldsymbol{\Sigma })=a(\bolds... ...r(\boldsymbol{y};\boldsymbol{<tex2html_comment_mark>84 \mu })\right\} \right] ,$

(15)

where $\boldsymbol{\Sigma }$ is a symmetric positive-definite $k\times k$ matrix,

is an increasing function satisfying

and $g^{\prime }(0)>0,$ and $r(\boldsymbol{y};\boldsymbol{\mu })$ is a suitably defined

-vector of residuals satisfying $r(\boldsymbol{\mu };\boldsymbol{\mu })=% \boldsymbol{0}$ for $\boldsymbol{\mu }\in \Omega ,$ see Jørgensen and Lauritzen [14]. An example is the multivariate normal distributions $% \mathrm{N}_{k}(\boldsymbol{\mu },\boldsymbol{\Sigma })$ , which is obtained for $r(\boldsymbol{y};\boldsymbol{\mu })=\boldsymbol{y}-\boldsymbol{\mu }$ and

the identity function. Further examples includes the multivariate Laplace and

distributions, along with the class of elliptically contoured distributions of Fang [3]. Further generalizations and examples may be found in Jørgensen and Lauritzen [14], see also Jørgensen and Rajeswaran [16].

Bibliography

1: Bar-Lev, S.K. and Enis, P. (1986). Reproducibility and natural exponential families with power variance functions. Ann. Statist. 14, 1507-1522.
2: Barndorff-Nielsen, O.E. and Jørgensen, B. (1991). Some parametric models on the simplex. J. Multivariate Anal. 39, 106-116.
3: Fang, K.-T. (1997). Elliptically contoured distributions. In Encyclopedia of Statistical Sciences, Update Vol. 1 (Eds. S. Kotz, C.B. Read and D.L. Banks), pp. 212-218. New York: Wiley.
4: Hastie, T. (1987). A closer look at the deviance. Amer. Statist. 41, 16-20.
5: Hougaard, P. (1986). Survival models for heterogeneous populations derived from stable distributions. Biometrika 73, 387-396.
6: Jensen, J.L. (1981). On the hyperboloid distribution. Scand. J. Statist. 8, 193-206.
7: Jørgensen, B. (1983). Maximum likelihood estimation and large-sample inference for generalized linear and nonlinear regression models. Biometrika 70, 19-28.
8: Jørgensen, B. (1986). Some properties of exponential dispersion models. Scand. J. Statist. 13, 187-197.
9: Jørgensen, B. (1987). Exponential dispersion models (with discussion). J. Roy. Statist. Soc. Ser. B 49, 127-162.
10: Jørgensen, B. (1987). Small-dispersion asymptotics. Braz. J. Probab. Statist. 1, 59-90.
11: Jørgensen, B. (1992). Exponential dispersion models and extensions: A review. Internat. Statist. Rev. 60, 5-20.
12: Jørgensen, B. (1997). Proper dispersion models (with discussion). Braz. J. Probab. Statist. 11, 89-140.
13: Jørgensen, B. (1997). The Theory of Dispersion Models. London: Chapman & Hall.
14: Jørgensen, B. and Lauritzen, S.L. (2000). Multivariate dispersion models. J. Multivar. Anal. 74, 267-281.
15: Jørgensen, B., Martínez, J.R. and Tsao, M. (1994). Asymptotic behaviour of the variance function. Scand. J. Statist. 21, 223-243.
16: Jørgensen, B. and Rajeswaran, J. (2005). A generalization of Hotelling's $T^{2}$ . Communications in Statistics--Theory and Methods 34, 2179-2195.
17: Jørgensen, B. and Souza, M.P. (1994). Fitting Tweedie's compound Poisson model to insurance claims data. Scand. Actuarial J.69-93.
18: Lee, M.-L.T. and Whitmore, G.A. (1993). Stochastic processes directed by randomized time. J. Appl. Probab. 30, 302-314.
19: Morris, C.N. (1981). Models for positive data with good convolution properties. Memo no. 8949. California: Rand Corporation.
20: Morris, C.N. (1982). Natural exponential families with quadratic variance functions. Ann. Statist. 10, 65-80.
21: Nelder, J.A. and Wedderburn, R.W.M. (1972). Generalized linear models. J. Roy. Statist. Soc. Ser. A 135, 370-384.
22: Renshaw, A.E. (1993). An application of exponential dispersion models in premium rating. Astin Bull. 23, 145-147.
23: Sweeting, T.J. (1981). Scale parameters: A Bayesian treatment. J. Roy. Statist. Soc. Ser. B 43, 333-338.
24: Tweedie, M.C.K. (1947). Functions of a statistical variate with given means, with special reference to Laplacian distributions. Proc. Cambridge Phil. Soc. 49, 41-49.
25: Tweedie, M.C.K. (1984). An index which distinguishes between some important exponential families. In Statistics: Applications and new directions. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference (eds. J.K. Ghosh and J. Roy), pp. 579-604. Calcutta: Indian Statistical Institute.

Dispersion Models is owned by Bent JÃ¸rgensen.

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The code that caused the bug has now been removed. Classification "62E10","60E05"

How to Cite This Entry

Bent JÃ¸rgensen. "Dispersion Models" (version 4). StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies. Freely available at http://statprob.com/encyclopedia/TweedieDistribution.html

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