Suppose $x_{1}$ is a single observation from a distribution with density $f$ over $[0,1]$. It is desired to test $H_{0}: f(x)=1$ against $H_{1}: f(x)=2 x$.

Let $\delta:[0,1] \rightarrow\{0,1\}$ define a test by $\delta\left(x_{1}\right)=i \Longleftrightarrow$ 'accept $H_{i}$ '. Let $\alpha_{i}(\delta)=P\left(\delta\left(x_{1}\right)=1-i \mid H_{i}\right)$. State the Neyman-Pearson lemma using this notation.

Let $\delta$ be the best test of size $0.10$. Find $\delta$ and $\alpha_{1}(\delta)$.

Consider now $\delta:[0,1] \rightarrow\{0,1, \star\}$ where $\delta\left(x_{1}\right)=\star$ means 'declare the test to be inconclusive'. Let $\gamma_{i}(\delta)=P\left(\delta(x)=\star \mid H_{i}\right)$. Given prior probabilities $\pi_{0}$ for $H_{0}$ and $\pi_{1}=1-\pi_{0}$ for $H_{1}$, and some $w_{0}, w_{1}$, let

$$\operatorname{cost}(\delta)=\pi_{0}\left(w_{0} \alpha_{0}(\delta)+\gamma_{0}(\delta)\right)+\pi_{1}\left(w_{1} \alpha_{1}(\delta)+\gamma_{1}(\delta)\right)$$

Let $\delta^{*}\left(x_{1}\right)=i \Longleftrightarrow x_{1} \in A_{i}$, where $A_{0}=[0,0.5), A_{\star}=[0.5,0.6), A_{1}=[0.6,1]$. Prove that for each value of $\pi_{0} \in(0,1)$ there exist $w_{0}, w_{1}$ (depending on $\left.\pi_{0}\right)$ such that $\operatorname{cost}\left(\delta^{*}\right)=\min _{\delta} \operatorname{cost}(\delta) .\left[\right.$ Hint $\left.: w_{0}=1+2(0.6)\left(\pi_{1} / \pi_{0}\right) .\right]$

Hence prove that if $\delta$ is any test for which

$$\alpha_{i}(\delta) \leqslant \alpha_{i}\left(\delta^{*}\right), \quad i=0,1$$

then $\gamma_{0}(\delta) \geqslant \gamma_{0}\left(\delta^{*}\right)$ and $\gamma_{1}(\delta) \geqslant \gamma_{1}\left(\delta^{*}\right)$.