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

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.