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October 2002 Normal, gamma and inverse-Gaussian are the only NEFs where the bilateral UMPU and GLR tests coincide
Shaul K. Bar-Lev, Daoud Bshouty, Gérard Letac
Ann. Statist. 30(5): 1524-1534 (October 2002). DOI: 10.1214/aos/1035844987

Abstract

Consider an NEF $F$ on the real line parametrized by $\theta \in \Theta $. Also let $\theta _0$ be a specified value of $\theta $. Consider the test of size $\alpha$ for a simple hypothesis $H_0\dvtx \theta =\theta _0$ versus two sided alternative $H_1\dvtx \theta \neq \theta _0$. A~UMPU test of size~$\alpha $ then exists for any given $\alpha$. 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.

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Shaul K. Bar-Lev. Daoud Bshouty. Gérard Letac. "Normal, gamma and inverse-Gaussian are the only NEFs where the bilateral UMPU and GLR tests coincide." Ann. Statist. 30 (5) 1524 - 1534, October 2002. https://doi.org/10.1214/aos/1035844987

Information

Published: October 2002
First available in Project Euclid: 28 October 2002

zbMATH: 1016.62055
MathSciNet: MR1936330
Digital Object Identifier: 10.1214/aos/1035844987

Subjects:
Primary: 62G10

Keywords: Generalized likelihood ratio test , natural exponential families , uniformly most powerful unbiased test , variance functions

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.30 • No. 5 • October 2002
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