8.1. Most Powerful Tests 473and, from Equation (8.1.5), we obtain
∫CL(θ′′)−∫AL(θ′′)≥1
k[∫C∩AcL(θ′)−∫A∩CcL(θ′)]. (8.1.6)
However,
∫C∩AcL(θ′)−∫A∩CcL(θ′)=∫C∩AcL(θ′)+∫C∩AL(θ′)−∫A∩CL(θ′)−∫A∩CcL(θ′)=∫CL(θ′)−∫AL(θ′)=α−α=0.If this result is substituted in inequality (8.1.6), we obtain the desired result,
∫CL(θ′′)−∫AL(θ′′)≥ 0.If the random variables are of the discrete type, the proof is the same with integra-
tion replaced by summation.
Remark 8.1.1.As stated in the theorem, conditions (a), (b), and (c) are sufficient
ones for regionC to be a best critical region of sizeα. However, they are also
necessary. We discuss this briefly. Suppose there is a regionAof sizeαthat does
not satisfy (a) and (b) and that is as powerful atθ=θ′′asC, which satisfies (a),
(b), and (c). Then expression (8.1.5) would be zero, since the power atθ′′usingAis
equal to that usingC. It can be proved that to have expression (8.1.5) equal zero,A
must be of the same form asC. As a matter of fact, in the continuous case,AandC
would essentially be the same region; that is, they could differ only by a set having
probability zero. However, in the discrete case, ifPH 0 [L(θ′)=kL(θ′′)] is positive,
AandCcould be different sets, but each would necessarily enjoy conditions (a),
(b), and (c) to be a best critical region of sizeα.
It would seem that a test should have the property that its power should never
fall below its significance level; otherwise, the probability of falsely rejectingH 0
(level) is higher than the probability of correctly rejectingH 0 (power). We say a
test having this property isunbiased, which we now formally define:
Definition 8.1.2.LetXbe a random variable which has pdf or pmff(x;θ),where
θ∈Ω. Consider the hypotheses given in expression (8.1.1). LetX′=(X 1 ,...,Xn)
denote a random sample onX. Consider a test with critical regionCand levelα.
We say that this test isunbiasedif
Pθ(X∈C)≥α,for allθ∈ω 1.