Page 438 - Probability and Statistical Inference

8. Tests of Hypotheses 415

Thus, one implements the MP level a test as follows:

where is the upper 100α% point of the distribution. We leave out the
power calculation as an exercise. !
Example 8.3.10 Suppose that X is an observable random variable with the
pdf f(x), x ∈ ℜ. We wish to test

That is, to decide between the simple null hypothesis which specifies that X is
distributed as N(0, ½) versus the simple alternative hypothesis which speci-
fies that X has the Cauchy distribution. In view of the Remark 8.3.1, the MP
test will have the following form:

Now, we want to determine the points x for which the function (1 + x )
2 -1
exp(x ) becomes “large”. Let us define a new function, g(y) = (1 + y) exp(y),
-1
2
0 < y < 8. We claim that g(y) is increasing in y. In order to verify this claim, it
will be simpler to consider h(y) = log(g(y)) = y – log(1 + y) instead. Note that
= y/(1 + y) > 0 for 0 < y < 8. That is, h(y) is increasing in y so that g(y)
is then increasing in y too, since the log operation is monotonically increasing.
Hence, the “large” values of (1 + x ) exp(x ) would correspond to the “large”
2 -1
2
values of x or equivalently, the “large” values of |x|. Thus, the MP test given
2
by (8.3.29) will have the following simple form:
This test must also have the size a, that is

so that we have where z stands for the upper 50α% point of the
α/2
standard normal distribution. With this choice of k, one would implement the
MP level a test given by (8.3.30). For example, if a = .05, then z = 1.96 so
α/2
that k ≈ 1.385929. The associated power calculation can be carried out as
follows: