A property of natural exponential families in \(\mathbb{R}^ n\) with simple quadratic variance functions
type: Item
instance of: scholarly article
title: A property of natural exponential families in \(\mathbb{R}^ n\) with simple quadratic variance functions
zbMATH Open document ID: 0886.62057
DOI: 10.1016/S0378-3758(97)00033-5
published in: Journal of Statistical Planning and Inference
publication date: 1997-11-13T00:00:00Z
review text: Suppose that \(F\) is a natural exponential family \(\{a(\theta) \exp (\theta x)f(x) dx\); \(\theta\in\Theta\}\) on \(\mathbb{R}\), which is absolutely continuous with quadratic variance \(V_F (m)\) defined on the mean domain \(M_F\). \textit{B. Jørgensen} et al. [Can. J. Stat. 17, No. 1, 1-8 (1989; Zbl 0676.62009)] showed that \[ V_F(m) \left(\int^\infty_{-\infty} a(\theta) \exp (\theta m)d \theta \right) f(m)=1 \quad \text{on } M_F. \] The density on \(\Theta\), proportional to \(a( \theta) \exp (\theta m)\) is the conjugate prior corresponding to \(F\). A similar result was also shown to hold in the discrete case. Furthermore, it was shown that this does not characterize the natural exponential families endowed with quadratic variances. In this note we extend this concept to natural exponential families in \(\mathbb{R}^n\) which possess a ``simple quadratic variance'' (a term to be made precise).
Mathematics Subject Classification ID: 62H05
zbMATH DE Number: 1086144
zbMATH Keywords: Laplace transform
zbMATH Keywords: simple quadratic variance functions
zbMATH Keywords: Wishart distributions
zbMATH Keywords: conjugate prior
zbMATH Keywords: natural exponential families
author: Vanamamalai Seshadri
MaRDI profile type: Publication
cites work: Q4291263
cites work: Conjugate priors for exponential families
cites work: Q3671491
cites work: Sur une propriété des familles exponentielles naturelles de variance quadratique
cites work: Q4729155
cites work: Natural exponential families with quadratic variance functions
description: scientific article; zbMATH DE number 1086144