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Let $X$ and $Y$ be independent random variables, distributed as Binomial($p, n$) and Binomial($p^2, m$), respectively. Does a UMP test (for fixed level $\alpha$) exist for: $H_0: p \leq p_0 \text{ vs. } H_1: p > p_0$?

In order to invoke the Karlin-Rubin theorem the monotone likelihood property is required, but it's not clear to me if we have that.

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I don't see how MLR is going to applied here since the minimal sufficient statistic is $(X, Y)$. – guy Dec 21 '12 at 4:22

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