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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.