## Abstract

Let F be a full natural exponential family on R which is generated by a self-decomposable probability distribution P. We provide a necessary and sufficient condition on P under which all other elements of F are also self-decomposable. Moreover, we show that if F is self-decomposable (i.e., composed of self-decomposable elements), then the exponential dispersion model generated by F shares the same property. The statistical importance of such results is linked to the fact that self-decomposable distributions are absolutely continuous and unimodal, thus providing potential exponential dispersion models for modeling data stemming from absolutely continuous and unimodal populations.

Original language | English |
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Pages (from-to) | 147-152 |

Number of pages | 6 |

Journal | Statistics and Probability Letters |

Volume | 13 |

Issue number | 2 |

DOIs | |

State | Published - 27 Jan 1992 |

## Keywords

- Self-decomposable distribution
- exponential dispersion model
- infinitely divisible distribution
- natural exponential family
- unimodal distribution

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty