Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

474 Optimal Tests of Hypotheses

As the next corollary shows, the best test given in Theorem 8.1.1 is an unbiased
test.

Corollary 8.1.1.As in Theorem 8.1.1, letCbe the critical region of the best test
ofH 0 :θ=θ′versusH 1 :θ=θ′′. Suppose the significance level of the test isα.
LetγC(θ′′)=Pθ′′[X∈C]denote the power of the test. Thenα≤γC(θ′′).

Proof: Consider the “unreasonable” test in which the data are ignored, but a
Bernoulli trial is performed which has probabilityαof success. If the trial ends
in success, we rejectH 0. The level of this test isα. Because the power of a test
is the probability of rejectingH 0 whenH 1 is true, the power of this unreasonable
test isαalso. ButCis the best critical region of sizeαand thus has power greater
than or equal to the power of the unreasonable test. That is,γC(θ′′)≥α,whichis
the desired result.

Another aspect of Theorem 8.1.1 to be emphasized is that if we takeCto be
the set of all pointsxwhich satisfy

L(θ′;x)

L(θ′′;x)

≤k, k > 0 ,

then, in accordance with the theorem,Cis a best critical region. This inequality

can frequently be expressed in one of the forms (wherec 1 andc 2 are constants)

u 1 (x;θ′,θ′′)≤c 1

or

u 2 (x;θ′,θ′′)≥c 2.

Suppose that it is the first form,u 1 ≤c 1 .Sinceθ′ andθ′′are given constants,
u 1 (X;θ′,θ′′) is a statistic; and if the pdf or pmf of this statistic can be found when
H 0 is true, then the significance level of the test ofH 0 againstH 1 can be determined
from this distribution. That is,

α=PH 0 [u 1 (X;θ′,θ′′)≤c 1 ].

Moreover, the test may be based on this statistic; for if the observed vector value
ofXisx, we rejectH 0 (acceptH 1 )ifu 1 (x)≤c 1.
A positive numberkdetermines a best critical regionC whosesizeisα =
PH 0 [X∈C] for that particulark. It may be that this value ofαis unsuitable for
the purpose at hand; that is, it is too large or too small. However, if there is a
statisticu 1 (X) as in the preceding paragraph, whose pdf or pmf can be determined
whenH 0 is true, we need not experiment with various values ofk to obtain a
desirable significance level. For if the distribution of the statistic is known, or can
be found, we may determinec 1 such thatPH 0 [u 1 (X)≤c 1 ] is a desirable significance
level.
An illustrative example follows.