Page 452 - Probability and Statistical Inference

8. Tests of Hypotheses 429

Note that ψ* corresponds to a level a test and hence the power associated
with ψ must be at least as large as the power associated with ψ* since ψ is
MP level a. Thus,

In other words, the MP level a test arising from the Neyman-Pearson Lemma
is in fact unbiased.
Remark 8.5.1 In the Section 8.5.1, we had shown that there was no
UMP level a test for testing a simple null against the two-sided alternative
hypotheses about the unknown mean of a normal population with its variance
known. But, there is a good test available for this problem. In Chapter 11, we
will derive the likelihood ratio test for the same hypotheses testing problem.
It would be shown that the level a likelihood ratio test would

This coincides with the customary two-tailed test which is routinely used in
practice. It will take quite some effort to verify that the test given by (8.5.10)
is indeed the UMPU level α test. This derivation is out of scope at the level of
this book. The readers nonetheless should be aware of this result.

8.6 Exercises and Complements

8.2.1 Suppose that X , X , X , X are iid random variables from the N(θ, 4)
1
2
3
4
population where θ(∈ ℜ) is the unknown parameter. We wish to test H : θ =
0
2 versus H : θ = 5. Consider the following tests:
1

Find Type I and Type II error probabilities for each test and compare the
tests.
8.2.2 Suppose that X , X , X , X are iid random variables from a popula-
3
4
1
2
+
tion with the exponential distribution having unknown mean θ (∈ ℜ ). We
wish to test H : θ = 4 versus H : θ = 8. Consider the following tests:
0 1