# Is this a proper use of the Karlin-Rubin UMP test theorem?

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.