470 Optimal Tests of Hypotheseslevelof the test is the probability of a Type I error; i.e.,α=max
θ∈ω 0
Pθ(X∈C). (8.1.3)Note thatPθ(X∈C) should be read as the probability thatX∈Cwhenθis the
true parameter. Subject to tests having sizeα, we select tests that minimize Type II
error or equivalently maximize the probability of rejectingH 0 whenθ∈ω 1. Recall
that thepower functionof a test is given by
γC(θ)=Pθ(X∈C); θ∈ω 1. (8.1.4)In Chapter 4, we gave examples of tests of hypotheses, while in Sections 6.3 and
6.4, we discussed tests based on maximum likelihood theory. In this chapter, we
want to construct best tests for certain situations.
We begin with testing a simple hypothesisH 0 against a simple alternativeH 1.
Letf(x;θ) denote the pdf or pmf of a random variableX,whereθ∈Ω={θ′,θ′′}.
Letω 0 ={θ′}andω 1 ={θ′′}.LetX′=(X 1 ,...,Xn) be a random sample from
the distribution ofX. We now define a best critical region (and hence a best test)
for testing the simple hypothesisH 0 against the alternative simple hypothesisH 1.
Definition 8.1.1.LetCdenote a subset of the sample space. Then we say thatC
is abest critical regionof sizeαfor testing the simple hypothesisH 0 :θ=θ′
against the alternative simple hypothesisH 1 :θ=θ′′if
(a) Pθ′[X∈C]=α.
(b) And for every subsetAof the sample space,
Pθ′[X∈A]=α⇒Pθ′′[X∈C]≥Pθ′′[X∈A].This definition states, in effect, the following: In general, there is a multiplicity
of subsetsAof the sample space such thatPθ′[X∈A]=α. Suppose that there
is one of these subsets, sayC, such that whenH 1 is true, the power of the test
associated withCis at least as great as the power of the test associated with every
otherA.ThenCis defined as a best critical region of sizeαfor testingH 0 against
H 1.
As Theorem 8.1.1 shows, there is a best test for this simple versus simple case.
But first, we offer a simple example examining this definition in some detail.
Example 8.1.1.Consider the one random variableXthat has a binomial distri-
bution withn=5andp=θ.Letf(x;θ) denote the pmf ofXand letH 0 :θ=^12
andH 1 :θ=^34. The following tabulation gives, at points of positive probability
density, the values off(x;^12 ),f(x;^34 ), and the ratiof(x;^12 )/f(x;^34 ).