## Abstract

Let μ be a positive measure defined on the product of two vector spaces E=E_{1}×E_{2}. Let F=F(μ) be a natural exponential family (NEF) generated by μ such that the projection of F on E_{1} constitutes a NEF on E_{1}. This property, called a cut on E_{1}, has been defined and characterized by Barndorff-Nielsen (Information and Exponential Families, Wiley, Chichester) and further developed by Barndorff-Nielsen and Koudou (Theory Probab. Appl. 40 (1995) 361). Their results can be used to conclude two properties of NEFs with cuts. The first stating that a NEF F has a cut on E_{1} if and only if for all random vectors (X,Y) on E_{1}×E_{2}, having a distribution in F, the regression curve of Y on X is linear. The second property states that the linearity of the scedastic curve of Y on X is a necessary condition for F to have a cut on E_{1}. These two properties of linearity of the regression and scedastic curves provide, in some situations, rather easily verifiable conditions for examining whether a NEF has a cut. Moreover, they are used to provide some interesting characterizations. In particular, some characterizations of the Gaussian and Poisson NEFs are obtained as special cases.

Original language | English |
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Pages (from-to) | 215-221 |

Number of pages | 7 |

Journal | Statistics and Probability Letters |

Volume | 63 |

Issue number | 2 |

DOIs | |

State | Published - 15 Jun 2003 |

## Keywords

- Cut
- Natural exponential family
- Regression curve
- Scedastic curve

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty