## Abstract

Consider an NEF F on the real line parametrized by θ ∈ Θ. Also let θ_{0} be a specified value of θ. Consider the test of size α for a simple hypothesis H_{0}:θ =θ _{0} versus two sided alternative H_{1}:θ≠θ _{0} A UMPU test of size α then exists for any given α. Suppose that F is continuous. Therefore the UMPU test is nonrandomized and then becomes comparable with the generalized likelihood ratio test (GLR). Under mild conditions we show that the two tests coincide iff F is either a normal or inverse Gaussian or gamma family. This provides a new global characterization of this set of NEFs. The proof involves a differential equation obtained by the cancelling of a determinant of order 6.

Original language | English |
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Pages (from-to) | 1524-1534 |

Number of pages | 11 |

Journal | Annals of Statistics |

Volume | 30 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2002 |

## Keywords

- Generalized likelihood ratio test
- Natural exponential families
- Uniformly most powerful unbiased test
- Variance functions

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty