Page 442 - Probability and Statistical Inference

8. Tests of Hypotheses 419

value σ (< σ ) and then verify that the MP level a test between σ and arbi-
0
1
0
trarily chosen σ will have the following form:
1
where is the lower 100α% point of the distribution. See, for ex-
ample, the Figure 8.4.1. Obviously this test does not depend on the specific
choice of σ (< σ ). Hence, the test given by (8.4.2) is UMP level a for testing
1 0
H : σ = σ versus H : σ < σ . !
0 0 1 0
Example 8.4.3 (Example 8.3.6 Continued) Suppose that X , ..., X are iid
n
1
Bernoulli(p) where p ∈ (0, 1) is the unknown parameter. With preassigned α ∈
(0, 1), we wish to obtain the UMP level α test for H : p = p versus H : p >
0
0
1
p where p is a number from the interval (0, 1). Now, fix a value p (> p ) and
0
0
1
0
then from (8.3.17) recall that the test function associated with the MP level a
test between p and arbitrarily chosen p will have the following form:
0 1
where k, a positive integer, and γ ∈ (0, 1) are chosen so that the test described
by (8.4.3) has size α. Obviously the test does not depend on the specific
choice of p (> p ). Hence, the test given by (8.4.3) is UMP level α for testing
1 0
H : p = p versus H : p > p . !
0 0 1 0
One can similarly argue that the MP level α tests derived in the Examples
8.3.2-8.3.5 are also the UMP level α tests for the same simple null hypothesis
versus the corresponding upper- or lower-sided composite alternative hy-
pothesis. Verifications of such claims are left out as exercises.
The p-value of a test:
In tests of hypothesis, one often reports the final result of the test, that is
whether one accepts H or H at a chosen level α. There is, however, another
1
0
way to report the final result in practice using the notion called the p-value.
We emphasize that the p-value is a fully data driven quantification of “error”
and its interpretation is sometimes difficult.
Consider the Example 8.4.1. Suppose that we have the observed value
for the sample mean and we have the calculated value
of the test statistic,
Now, we may ask the following question:
What is the probability of observing a data which is similar to
the one on hand or more extreme, if H happens to be true?
0
This probability will be called the p-value.