Abstract
Consider an NEF F on the real line parametrized by θ ∈ Θ. Also let θ0 be a specified value of θ. Consider the test of size α for a simple hypothesis H0:θ =θ 0 versus two sided alternative H1: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