Normal, gamma and inverse-Gaussian are the only NEFs where the bilateral UMPU and GLR tests coincide

• 成果类型: Article
• 署名作者: Bar-Lev, SK; Bshouty, D; Letac, G
• 署名单位: University of Haifa; Technion Israel Institute of Technology; Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 2002
• 页码: 1524-1534
• 关键词: exponential-families; variance functions

摘要

Consider an NEF F on the real line parametrized by theta is an element of Theta. Also let 00 be a specified value of theta. Consider the test of size alpha for a simple hypothesis H-0:theta = theta(0) versus two sided alternative H-1:theta not equal 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.