Unbiased and Consistent Tests

In the case of estimation of parameters, it was necessary to define some desirable properties for estimators, to enable us to have criteria for choosing between competing estimators. Similarly in hypothesis testing, we would like to use a test that is ``best'' in some sense. Note that a test specifies a critical region. Alternatively, the choice of a critical region defines a test. That is, the terms `test' and `critical region' can, in this sense, be used interchangeably. So if we define a best critical region, we have defined a best test.

The analogue for unbiasedness and consistency in estimation are defined below for hypothesis testing.

Definition 9..3
A test is unbiased if $P_{\theta}$(rejecting $H_0\vert H_1$) is always greater than $P_{\theta}$(rejecting $H_0\vert H_0$). That is,

$$\min_{\theta \in H_1}\pi(\theta) \geq \max_{\theta \in H_0}\pi(\theta).$$

Definition 9..4
A sequence of tests $\{\psi_n\}$, each of size $\alpha$, is consistent if their power functions approach 1 for all $\theta$ specified by the alternative. That is,

$$\pi_{\psi_n}(\theta) \rightarrow 1, \mbox{ for } \theta \in H_1.$$