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STAT 801: Mathematical Statistics

Hypothesis Testing and Decision Theory

Decision analysis of hypothesis testing takes $D=\{0,1\}$ and

$$L(d,\theta)=1($$make an error$$)$$

or more generally $L(0,\theta) = \ell_1 1(\theta\in\Theta_1)$ and $L(1,\theta) = \ell_2 1(\theta\in\Theta_0)$ for two positive constants $\ell_1$ and $\ell_2$. We make the decision space convex by allowing a decision to be a probability measure on $D$. Any such measure can be specified by $\delta=P($reject$)$ so ${\cal D} =[0,1]$. The loss function of $\delta\in[0,1]$ is

$$L(\delta,\theta) = (1-\delta)\ell_1 1(\theta\in\Theta_1) + \delta \ell_0 1(\theta\in\Theta_0) \, .$$

Simple hypotheses: Prior is $\pi_0>0$ and $\pi_1>0$ with $\pi_0+\pi_1=1$.

Procedure: map from sample space to $\cal D$ - a test function.

Risk function of procedure $\phi(X)$ is a pair of numbers:

$$R_\phi(\theta_0) = E_0(L(\delta,\theta_0))$$

and

$$R_\phi(\theta_1) = E_1(L(\delta,\theta_1))$$

We find

$$R_\phi(\theta_0) = \ell_0 E_0(\phi(X)) =\ell_0\alpha$$

and

$$R_\phi(\theta_1) = \ell_1 E_1(1-\phi(X)) = \ell_1 \beta$$

The Bayes risk of $\phi$ is

$$\pi_0\ell_0 \alpha+ \pi_1\ell_1\beta$$

We saw in the hypothesis testing section that this is minimized by

$$\phi(X) = 1(f_1(X)/f_0(X) > \pi_0\ell_0/(\pi_1\ell_1))$$

which is a likelihood ratio test. These tests are Bayes and admissible. The risk is constant if $\beta\ell_1 = \alpha\ell_0$; you can use this to find the minimax test in this context.