Hi I am working on the following problem
A single observation $X$ is made from one of three densities listed below with parameter space $\Theta=\{0,1,2\}$. \begin{align*} x=0\hspace{0.4cm}x=1\hspace{.4cm}x=2\hspace{.4cm}x=3\hspace{.4cm}x=4\\ f(x|\theta=0)\,\,\,\,\,\, 0.05\hspace{0.9cm} 0.05\hspace{0.9cm} 0.40\hspace{0.9cm} 0.50\hspace{0.9cm}0.00\\ f(x|\theta=1)\,\,\,\,\,\, 0.30\hspace{0.9cm} 0.40\hspace{0.9cm} 0.05\hspace{0.9cm} 0.20\hspace{0.9cm}0.05\\ f(x|\theta=2)\,\,\,\,\,\, 0.40\hspace{0.9cm} 0.30\hspace{0.9cm} 0.10\hspace{0.9cm} 0.10\hspace{0.9cm}0.10 \end{align*}
a) Find the likelihood ratio test of size $\alpha=0.1$ for testing $H_0: \theta=0$ against $H_1: \theta=\{1,2\}$
b) Is the test in part a UMP against the alternative? Why or why not?
c) Determine whether there exists a UMP test of size $\alpha=0.05$ for testing $H_0: \theta=0$ against $H_1: \theta=\{1,2\}$.
I got the part (a) which is \begin{align*} R=\{x\in\{0,1,4\}\}\\ \alpha=P_{\theta=0}(x\in\{0,1,4\}) \end{align*}
I am stuck with (b) and (c) any help would be highly appreciated. Thanks in advance.
Hints, in something approaching plain English. I hope you will learn something by expanding them carefully to give details and use formal notation as in your text:
In (a) there is really only one LR test at level 10%, and you have found it. Thus "uniformly" is easily satisfied in (b).
However in (c), there are two LR tests at level 5%, having rejection regions $\{0,4\}$ and $\{1,4\}$. But respective powers against $\theta = 1$ and $2$ are .35 and .45 for $\{0,4\},$ and .50 and .40 for $\{1,4\}$. So one rejection region is better for $\theta = 1$ and the other is better for $\theta = 2,$ and that's not "uniformly" most powerful.