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For iid $X_1,...,X_n$ and the unknown parameter $\theta>1$, suppose that the likelihood function of a particular sample is given by: $$L(x;\theta)=log(\theta)^n\theta^{{n-\sum_{{i=1}}^nx_i}} I(x_{(1)}>1), $$ where $T(X)=\sum_{i=1}^nX_i$ is sufficient for $\theta$. My task is to find the UMP level-$\alpha$ test for $H_0$: $\mu\leq \mu_0$ vs $H_1$: $\mu>\mu_0$, where $\mu=E[X]=1+\frac{1}{log\theta}$. Here's my work:

First, I rewrite the likelihood function in terms of $\mu$ and $T$, and get the following: $$L(t;\mu)=(\mu-1)^{-n}e^{\frac{n-t}{\mu-1}}. $$

To use Karlin-Rubin, we must have that the likelihood function is non-decreasing in $T$, the sufficient statistic. Since the likelihood function in this question is actually non-increasing, can I simply use $-T$ as my sufficient statistic and then use the Karlin-Rubin theorem?

Edit: If this is not an appropriate use of Karlin-Rubin, I would appreciate a suggestion for finding the UMP test.

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