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

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. [References: 6]