Testing Statistical Hypotheses

Let $X$ be a sample of size 1. Let $H_0: X \sim U[0,1], \; H_1: X \sim \mathrm{Exp}(1)$. Construct the most powerful test of significance level $\alpha \in (0,1)$ to distinguish between these hypotheses, and compute its power.

$X_1, \dots , X_n$ is a sample from $\mathrm{Exp}(\theta )$. Construct the UMP test for testing $H_0: \theta = \theta_0$ against

$X_1, \dots , X_n$ is a sample from $N(\theta , 1)$. Construct the UMP test for

$X_1, \dots , X_n$ is a sample from $\mathrm{Bern}(\theta )$. Prove that there is no UMP test of arbitrary significance level $\alpha$ for testing the hypothesis $H_0: \theta = \theta_0 \in (0,1)$ against $H_1: \theta \neq \theta_0$.

$X_1, \dots , X_n$ is a sample from $N(0, \theta )$. Construct the UMP test of significance level $\alpha$ for testing $H_0: \theta = \theta_0$ against $H_1: \theta = \theta_1$, and find its power.

$X_1, \dots , X_n$ is a sample from $U[0,\theta ], \; \theta > 0$. Construct the UMP test of significance level $\alpha$ for testing $H_0: \theta = \theta_0$ against $H_1: \theta \neq \theta_0$ in the form

Let $\left\{ \mathbb {P}_\theta , \theta \in \Theta \right\}$ be a family with a nonincreasing likelihood ratio in the statistic $T(X)$, and let $\alpha < 1$ be some positive number. Construct the UMP test of significance level $\alpha$ for testing the hypothesis $H_0$ against the alternative $H_1$, where

Show that any uniformly most powerful unbiased (i.e., $\inf_{\theta \in \Theta_0} \beta (\theta , S) \geq \sup_{\theta \in \Theta_1} \beta (\theta , S)$) test $S$ is admissible, i.e., there is no other test $R$ that is at least as powerful as $S$ for all alternatives and strictly more powerful for at least one alternative.