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