## Abstract

Let ℱ be a natural exponential family onℝ and (V, Ω) be its variance function. Here, Ω is the mean domain of ℱ and V, defined on Ω, is the variance of ℱ. A problem of increasing interest in the literature is the following: Given an open interval Ω⊂ℝ and a function V defined on Ω, is the pair (V, Ω) a variance function of some natural exponential family? Here, we consider the case where V is a polynomial. We develop a complex-analytic approach to this problem and provide necessary conditions for (V, Ω) to be such a variance function. These conditions are also sufficient for the class of third degree polynomials and certain subclasses of polynomials of higher degree.

Original language | English |
---|---|

Pages (from-to) | 69-82 |

Number of pages | 14 |

Journal | Probability Theory and Related Fields |

Volume | 94 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1992 |

## Keywords

- Mathematics Subject Classification (1980): 62E10, 60J30

## ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty