Page 435 - Probability and Statistical Inference

412 8. Tests of Hypotheses

that is, we will reject the null hypothesis H if
0

Now since λ > λ , the “large values” of the lhs in (8.3.20) correspond to the
1 0
“large values” of Hence, the MP test defined by (8.3.20) can be
rephrased as:

We then write down the test function as follows:

where a positive integer k and γ ∈ (0, 1) are to be chosen in such a way that
the test has the size α. Observe that has Poisson(nλ ) distribution
0
under H . First, we determine the smallest integer value of k such that
0
and let

where

Now, with k and γ defined by (8.3.22), one can check that the Type I error
probability is

Thus, we have the MP level a test. If = k, then one would employ
appropriate randomization and reject H with probability γ. The following
0
table provides some values of k, γ for specific choices of n and α. The reader
Table 8.3.3. Values of k and γ in the Poisson Case

n = 10 α = .10 n = 10 α = .05
λ k γ λ k γ
0 0
.15 3 .274 .15 4 .668
.30 5 .160 .35 7 .604

n = 20 α = .05 n = 25 α = .10
λ k γ λ k γ
0 0
.40 13 .534 .28 10 .021
.50 15 .037 .40 14 .317