Exponential dispersion models. (English) Zbl 0662.62078

Jørgensen, Bent

J. R. Stat. Soc., Ser. B 49, 127-162 (1987).

A multidimensional extension of the generalized linear model of P. MacCullagh and J. A. Nelder [Generalized linear models. (1983; Zbl 0588.62104)] is considered. These models, called exponential dispersion models, are defined by a family of probability measures denoted \(P_{\lambda,\theta}\) such that there exists a measure \(P_{\lambda}\) with respect to which the density is \[ dP_{\lambda,\theta}/dP_{\lambda}=e^{\lambda \{y^{\tau}\theta - k(\theta)\}},\quad y\in {\mathbb{R}}^ k,\quad and\quad (\lambda,\theta)\in \Lambda \times \Theta. \] The asymptotic properties of a weighted sum of such independent random variables are studied. The problems of inference in the case where \(\lambda\) is known or unknown are considered. The links with the model of Nelder and McCullagh are studied. Many examples are given and there is a large discussion given by 15 discussants.

Reviewer: J.-R.Mathieu

MSC:

62J99	Linear inference, regression
62E20	Asymptotic distribution theory in statistics
62H12	Estimation in multivariate analysis
62H15	Hypothesis testing in multivariate analysis

Keywords:

moment generating function; asymptotic normality; Poisson; binomial; negative binomial; normal; gamma; inverse Gaussian distributions; combinations; compound distributions; convolution; exponential families; longitudinal data; mixtures; power variance functions; small-dispersion asymptotics; stable distribution; variance components; variance functions; generalized linear model; exponential dispersion models; examples

Citations:

Zbl 0588.62104