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Exponential dispersion models. (English) Zbl 0662.62078
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