Minimax Procedures

Definition

We want to test $H_0:\theta ={\theta }^{\prime },H_1:\theta ={\theta }^{\prime \prime }$ Let $\mathcal{L}(\theta ,\delta )$ be a Loss Function such that $\mathcal{L}({\theta }^{\prime },{\theta }^{\prime })=\mathcal{L}({\theta }^{\prime \prime },{\theta }^{\prime \prime })=0$ and $\mathcal{L}({\theta }^{\prime },{\theta }^{\prime \prime })>0,\mathcal{L}({\theta }^{\prime \prime },{\theta }^{\prime })>0$, $C$ be a rejection region, $C^{\complement }$ be an acceptance region, and $R(\theta ,\delta )=E[\mathcal{L}(\theta ,\delta )]$ be a Risk Function Then, $R({\theta }^{\prime },C)=\alpha \mathcal{L}({\theta }^{\prime },{\theta }^{\prime \prime }),R({\theta }^{\prime \prime },C)=\beta \mathcal{L}({\theta }^{\prime \prime },{\theta }^{\prime })$ The minimax procedure is to find $C$ such that $\max \{R({\theta }^{\prime },C),R({\theta }^{\prime \prime },C)\}$ is minimized

The minimax solution is the region $C=\left\{(x_1,x_2,\dots ,x_n)|\frac{L({\theta }^{\prime })}{L({\theta }^{\prime \prime })}\le k\right\}$, where $k$ satisfies $R({\theta }^{\prime },C)=R({\theta }^{\prime \prime },C)$