## Abstract

A natural exponential family (NEF)F in ℝ^{n}, n>1, is said to be diagonal if there exist n functions, a_{1},..., a_{n}, on some intervals of ℝ, such that the covariance matrix V_{F}(m) of F has diagonal (a_{1}(m_{1}),..., a_{n}(m_{n})), for all m=(m_{1},..., m_{n}) in the mean domain of F. The family F is also said to be irreducible if it is not the product of two independent NEFs in ℝ^{k} and ℝ^{n-k}, for some k=1,..., n-1. This paper shows that there are only six types of irreducible diagonal NEFs in ℝ^{n}, that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: If F is an NEF in ℝ^{n}, under what conditions is its projection p(F) in ℝ^{k}, under p(x_{1},..., x_{n}):=(x_{1},..., x_{k}), k=1,..., n-1, still an NEF in ℝ^{k}? The answer turns out to be rather predictable. It is the case if, and only if, the principal k×k submatrix of V_{F}(m_{1},..., m_{n}) does not depend on (m_{k+1},..., m_{n}).

Original language | English |
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Pages (from-to) | 883-929 |

Number of pages | 47 |

Journal | Journal of Theoretical Probability |

Volume | 7 |

Issue number | 4 |

DOIs | |

State | Published - Oct 1994 |

## Keywords

- Gamma distribution
- Jørgensen set
- Lancaster sequences
- Poisson distribution
- hyperbolic secant distribution
- multinomial distribution
- natural exponential families
- negative multinomial distribution
- normal distribution
- variance function

## ASJC Scopus subject areas

- Statistics and Probability
- General Mathematics
- Statistics, Probability and Uncertainty