I am posting this question as a solution checking.
Let $X_1,...,X_{30}$ be a random sample from the exponential distribution with unknown mean $\mu\in \{1,1/\delta\}$ (where $\delta>1)$. Consider the best test of $H_0:\mu=1$ v.s. $H_1:\mu=1/\delta$ with power given by $\beta=\int_0^{17}\dfrac {\delta^{30}}{\Gamma(30)}y^{29}e^{-\delta y} dy$. What is the critical region of the test ?
My work: By Neyman-Pearson lemma, the best test is given by $L(1;\mathbb x)/L(1/\delta;\mathbb x)\le k$ i.e. $e^{-\sum x_i}/e^{-\delta \sum x_i} \le c$ i.e. $e^{(\delta-1)\sum x_i}\le c $ i.e. $\sum x_i \le c_1$ for some constant $c_1$. Now given $P_{H_1}(\sum_{i=1}^{30} X_i \le c_1)=\int_0^{17}\dfrac {\delta^{30}}{\Gamma(30)}y^{29}e^{-\delta y} dy$. Now under $H_1$, $X_i \sim exp (\delta)$ , so $\sum_{i=1}^{30} X_i \sim Gamma(30,1/\delta)$, hence $c_1=17$.
Thus the critical region is $\sum_{i=1}^{30} X_i\le 17$.
Am I correct ?
Please let me know.
Thanks in advance
