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.
Publié le : 2002-10-14
Classification:
Generalized likelihood ratio test,
uniformly most powerful unbiased test,
natural exponential families,
variance functions,
62G10
@article{1035844987,
author = {Bar-Lev, Shaul K. and Bshouty, Daoud and Letac, G\'erard},
title = {Normal, gamma and inverse-Gaussian are the only NEFs where the bilateral UMPU and GLR tests coincide},
journal = {Ann. Statist.},
volume = {30},
number = {1},
year = {2002},
pages = { 1524-1534},
language = {en},
url = {http://dml.mathdoc.fr/item/1035844987}
}
Bar-Lev, Shaul K.; Bshouty, Daoud; Letac, Gérard. Normal, gamma and inverse-Gaussian are the only NEFs where the bilateral UMPU and GLR tests coincide. Ann. Statist., Tome 30 (2002) no. 1, pp. 1524-1534. http://gdmltest.u-ga.fr/item/1035844987/