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Let $X_1,X_2,\dots,X_n$ be iid Beta($\theta,\theta$) samples. Is there a UMP level $\alpha$ test of $H_0:\theta=1$ vs $H_A:\theta\neq1$?

We first test $H_0:\theta=1$ vs $H_A:\theta<1$. The likelihood ratio is $$\lambda(X)=\frac{L(X|H_0)}{L(X|H_A)}=C\prod_{i=1}^{n}\frac{1}{x_i^{\theta_A-1}(1-x_i)^{\theta_A-1}}=C\exp\left(\sum_{i=1}^{n}(1-\theta_A)\log(x_i(1-x_i))\right)$$ where $C$ is a constant. $\lambda(X)$ is an convex function.

To test $H_0:\theta=1$ vs $H_A:\theta>1$. The likelihood ratio is $$\lambda(X)=\frac{L(X|H_0)}{L(X|H_A)}=C\prod_{i=1}^{n}\frac{1}{x_i^{\theta_A-1}(1-x_i)^{\theta_A-1}}=C\exp\left(\sum_{i=1}^{n}(-\theta_A+1)\log(x_i(1-x_i))\right)$$ where $C$ is a constant. $\lambda(X)$ is an concave function.

We reject different values in these two cases. There is no UMP test.

I'm not sure about my solution. Any suggestions and comments are welcome.

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