A property of natural exponential families in \(\mathbb{R}^ n\) with simple quadratic variance functions (Q1372418)

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).