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 $H_1:\theta\neq\theta_0$.

Tags : statistics

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