Page 434 - Probability and Statistical Inference

8. Tests of Hypotheses 411

where one has

Now, with k and γ defined by (8.3.18), one can check that the Type I error
probability is

Thus, we have the MP level α test. If then one rejects H with
0
probability γ. For example, if γ = .135, then consider three-digit random num-
bers 000, 001, ..., 134, 135, ..., 999 and look at a random number table to
draw one three-digit number. If we come up with one of the numbers 000,
001, ... or 134, then and only then H will be rejected. This is what is known
0
as randomization. The following table provides the values of k and γ for some
specific choices of n and α.
Table 8.3.2. Values of k and γ in the Bernoulli Case

n = 10 α = .10 n = 10 α = .05
p k γ p k γ
0 0
.2 4 .763 .2 4 .195
.4 6 .406 .6 8 030

n = 20 α = .05 n = 25 α = .10
p k γ p k γ
0 0
.3 9 .031 .5 16 .757
.5 14 .792 .6 18 .330

The reader should verify some of the entries in this table by using the expres-
sions given in (8.3.19). !
+
Example 8.3.7 Suppose that X , ..., X are iid Poisson(λ) where λ ∈ ℜ is
1
n
the unknown parameter. With preassigned α ∈ (0, 1), we wish to derive the
MP level α test for H : λ = λ versus H : λ = λ (> λ ) where λ , λ are two
0
1
0
1
1
0
0
positive numbers. Both H and H are simple hypothesis and the Neyman-
0
1
Pearson Lemma applies Then, writing the likeli-
hood function is given by
The MP test will have the following form: