# No UMP test for $H_0:\theta=\theta_0$ vs $H_1:\theta\neq\theta_0$ for exponential family

by Oliver Diaz   Last Updated September 12, 2019 04:20 AM

Suppose $$\{f_\theta(x)=C(\theta)e^{G(\theta)T(x)}\}$$ is a family of probability densities with respect to a $$\sigma$$--finite measure $$\mu$$ on $$(\mathbb{R},\mathscr{B}(\mathbb{R})$$. Further, assume that $$G(\theta)$$ is strictly increasing.

The problem is to show that there is no uniform most powerful test for testing $$H_0:\theta=\theta_0$$ vs $$H_1:\theta\neq\theta_0$$.

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