Page 431 - Probability and Statistical Inference

408 8. Tests of Hypotheses

variables, so that are iid . Hence, is distributed as
when H is true. This test must also be size a. Let us equivalently rewrite the
0
same test as follows:

where is the upper 100a% point of the distribution. See, for ex-
ample, the Figure 8.3.4. The test from (8.3.11) asks us to reject H for large
0
enough values of whereas (8.3.12) asks us to reject H for large
0
enough values of We call the associated test statis-
tic. The order of “largeness”, that is the choice of k, depends on the normal-
ized test statistic and its distribution under H . Under H , since is
0
0
distributed as a random variable, we have:

Thus, we have the MP level a test. Here, the critical region R = {X =
!
Example 8.3.4 Suppose that X , ..., X are iid having the Uniform(0, θ)
1
n
distribution with the unknown parameter θ (> 0). With preassigned α ∈ (0,
1), we wish to obtain the MP level α test for H : θ = θ versus H : θ = θ (>
0
0
1
1
θ ) where θ , θ are two positive numbers. Both H , H are simple hypothesis
0
0
1
0
1
and the Neyman-Pearson Lemma applies. The likelihood function is given by
In view of the Remark 8.3.1, the MP test will have the following form:
that is, we will reject the null hypothesis H if and only if
0

Note that the MP test given by (8.3.13) is not in the implementable form.
n
Under H , the pdf of the statistic T = X is given by nt /θ . Hence we can
n-1
0
n:n
0
determine k as follows: