Chapter 8
Optimal Tests of Hypotheses
8.1 MostPowerfulTests
In Section 4.5, we introduced the concept of hypotheses testing and followed it with
the introduction of likelihood ratio tests in Chapter 6. In this chapter, we discuss
certain best tests.
For convenience to the reader, in the next several paragraphs we quickly review
concepts of testing that were presented in Section 4.5. We are interested in a random
variableXthat has pdf or pmff(x;θ), whereθ∈Ω. We assume thatθ∈ω 0 or
θ∈ω 1 ,whereω 0 andω 1 are disjoint subsets of Ω andω 0 ∪ω 1 =Ω. Welabelthe
hypotheses as
H 0 : θ∈ω 0 versusH 1 : θ∈ω 1. (8.1.1)
The hypothesisH 0 is referred to as thenull hypothesis, whileH 1 is referred to
as thealternative hypothesis.ThetestofH 0 versusH 1 isbasedonasample
X 1 ,...,Xnfrom the distribution ofX. In this chapter, we often use the vector
X′=(X 1 ,...,Xn) to denote the random sample andx′=(x 1 ,...,xn)todenote
the values of the sample. LetSdenote the support of the random sampleX′=
(X 1 ,...,Xn).
AtestofH 0 versusH 1 is based on a subsetCofS.ThissetCis called the
critical regionand its corresponding decision rule is
RejectH 0 (AcceptH 1 )ifX∈C (8.1.2)
RetainH 0 (RejectH 1 )ifX∈Cc.Note that a test is defined by its critical region. Conversely, a critical region defines
atest.
Recall that the 2×2 decision table, Table 4.5.1, summarizes the results of the
hypothesis test in terms of the true state of nature. Besides the correct decisions,
two errors can occur. AType Ierror occurs ifH 0 is rejected when it is true, while
aType IIerror occurs ifH 0 is accepted whenH 1 is true. Thesizeorsignificance
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