Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

508 Optimal Tests of Hypotheses

random variablesX 1 ,X 2 ,...,Xndepend upon the parameterθ.Herenis a fixed
positive integer. This pdf is denoted byL(θ;x 1 ,x 2 ,...,xn) or, for brevity, byL(θ).
Letθ′andθ′′bedistinctandfixedvaluesofθ. We wish to test the simple hypothesis
H 0 :θ=θ′against the simple hypothesisH 1 :θ=θ′′. Thus the parameter space is
Ω={θ:θ=θ′,θ′′}. In accordance with the decision function procedure, we need
a functionδof the observed values ofX 1 ,...,Xn(or, of the observed value of a
statisticY) that decides which of the two values ofθ,θ′orθ′′, to accept. That is,
the functionδselects eitherH 0 :θ=θ′orH 1 :θ=θ′′. We denote these decisions
byδ=θ′andδ=θ′′, respectively. LetL(θ, δ) represent the loss function associated
with this decision problem. Because the pairs (θ=θ′,δ=θ′)and(θ=θ′′,δ=θ′′)
represent correct decisions, we shall always takeL(θ′,θ′)=L(θ′′,θ′′)=0. Onthe
other hand, if eitherδ=θ′′whenθ=θ′orδ=θ′whenθ=θ′′,thenapositivevalue
should be assigned to the loss function; that is,L(θ′,θ′′)>0andL(θ′′,θ′)>0.
It has previously been emphasized that a test ofH 0 :θ=θ′againstH 1 :θ=θ′′
can be described in terms of a critical region in the sample space. We can do the
same kind of thing with the decision function. That is, we can choose a subset ofC
of the sample space and if (x 1 ,x 2 ,...,xn)∈C, we can make the decisionδ=θ′′;
whereas if (x 1 ,x 2 ,...,xn)∈Cc, the complement ofC, we make the decisionδ=θ′.
Thus a given critical regionCdetermines the decision function. In this sense, we
may denote the risk function byR(θ, C) instead ofR(θ, δ). That is, in a notation
used in Section 7.1,

R(θ, C)=R(θ, δ)=

∫

C∪Cc

L(θ, δ)L(θ).

Sinceδ=θ′′if (x 1 ,...,xn)∈Candδ=θ′if (x 1 ,...,xn)∈Cc,wehave

R(θ, C)=

∫

C

L(θ, θ′′)L(θ)+

∫

Cc

L(θ, θ′)L(θ). (8.5.1)

If, in Equation (8.5.1), we takeθ=θ′,thenL(θ′,θ′) = 0 and hence

R(θ′,C)=

∫

C

L(θ′,θ′′)L(θ′)=L(θ′,θ′′)

∫

C

L(θ′).

On the other hand, if in Equation (8.5.1) we letθ=θ′′,thenL(θ′′,θ′′) = 0 and,
accordingly,

R(θ′′,C)=

∫

Cc

L(θ′′,θ′)L(θ′′)=L(θ′′,θ′)

∫

Cc

L(θ′′).

It is enlightening to note that ifγ(θ) is the power function of the test associated
with the critical regionC,then

R(θ′,C)=L(θ′,θ′′)γ(θ′)=L(θ′,θ′′)α,

whereα=γ(θ′) is the significance level; and

R(θ′′,C)=L(θ′′,θ′)[1−γ(θ′′)] =L(θ′′,θ′)β,