Page 437 - Probability and Statistical Inference

414 8. Tests of Hypotheses

But, f (x)/f (x) = 4x -3/2 and hence the test given in (8.3.26) can be rewritten
1
0
as:

This test must also have the size a, that is we require:

so that k = 4α . With k = 4α , one would implement the MP level α test
1/3
1/3
given by (8.3.27). The associated power calculation can be carried out as
follows:

when k = 4α . When a = .05, .01 and .001, the power is respectively .223606,
1/3
.1 and .031623. !
Example 8.3.9 (Example 8.3.8 Continued) Suppose that X and X are
1
2
observable random variables with the common pdf given by f(x), x ∈ ℜ.
Consider the two functions f (x) and f (x) defined in the Example 8.3.8. As in
0
1
the earlier example, we wish to determine the MP level α test for
The reader should check that the MP test has the following form:

Let us write F (x) for the distribution function which corresponds to the pdf
0
f (x) so that
0

Under H , it is known that -2log{F (X )}, i = 1, 2, is distributed as iid One
0 0 i
may go back to the Example 4.2.5 in this context. The test defined via (8.3.28)
must also have size a, that is, we require: