Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

8.2. Uniformly Most Powerful Tests 479

8.1.10.LetX 1 ,X 2 ,...,X 10 denote a random sample of size 10 from a Poisson
distribution with meanθ. Show that the critical regionCdefined by

∑ 10
1 xi≥^3
is a best critical region for testingH 0 :θ=0.1 againstH 1 :θ=0.5. Determine,
for this test, the significance levelαand the power atθ=0.5. Use the R function
ppois.

8.2 UniformlyMostPowerfulTests

This section takes up the problem of a test of a simple hypothesisH 0 against an
alternative composite hypothesisH 1. We begin with an example.

Example 8.2.1.Consider the pdf

f(x;θ)=

{ 1

θe

−x/θ 0 <x<∞
0elsewhere
of Exercises 8.1.2 and 8.1.3. It is desired to test the simple hypothesisH 0 :θ=2
against the alternative composite hypothesisH 1 :θ>2. Thus Ω ={θ:θ≥ 2 }.
A random sample,X 1 ,X 2 ,ofsizen= 2 is used, and the critical region isC =
{(x 1 ,x 2 ):9. 5 ≤x 1 +x 2 <∞}. It was shown in the exercises cited that the
significance level of the test is approximately 0.05 and the power of the test when
θ= 4 is approximately 0.31. The power functionγ(θ)ofthetestforallθ≥2is

γ(θ)=1−

∫ 9. 5

0

∫ 9. 5 −x 2

0

1

θ^2

exp

(

−

x 1 +x 2

θ

)

dx 1 dx 2

=

(

θ+9. 5

θ

)

e−^9.^5 /θ, 2 ≤θ.

For example,γ(2) = 0.05,γ(4) = 0.31, andγ(9.5) = 2/e≈ 0 .74. It is shown
(Exercise 8.1.3) that the setC={(x 1 ,x 2 ):9. 5 ≤x 1 +x 2 <∞}is a best critical
region of size 0.05 for testing the simple hypothesisH 0 :θ= 2 against each simple
hypothesis in the composite hypothesisH 1 :θ>2.
The preceding example affords an illustration of a test of a simple hypothesis
H 0 that is a best test ofH 0 against every simple hypothesis in the alternative
composite hypothesisH 1. We now define a critical region, when it exists, which
is a best critical region for testing a simple hypothesisH 0 against an alternative
composite hypothesisH 1. It seems desirable that this critical region should be a
best critical region for testingH 0 against each simple hypothesis inH 1 .Thatis,
the power function of the test that corresponds to this critical region should be at
least as great as the power function of any other test with the same significance
level for every simple hypothesis inH 1.

Definition 8.2.1.The critical regionCis auniformly most powerful (UMP)
critical regionof sizeαfor testing the simple hypothesisH 0 against an alternative
composite hypothesisH 1 if the setCis a best critical region of sizeαfor testing
H 0 against each simple hypothesis inH 1. A test defined by this critical regionC
is called auniformly most powerful (UMP) test, with significance levelα,for
testing the simple hypothesisH 0 against the alternative composite hypothesisH 1.