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402    8. Tests of Hypotheses

                                 is choosen small. In practice, one often chooses α = .10, .05 or .01 unless
                                 otherwise stated. But the experimenter is free to choose any appropriate a
                                 value.
                                    Note that L(x; θ ), i = 0, 1, are two completely specified likelihood func-
                                                  i
                                 tions. Intuitively speaking, a test for H  versus H  comes down to the com-
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                                                                           1
                                 parison of L(x; θ ) with L(x; θ ) and figure out which one is significantly
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                                 larger. We favor the hypothesis associated with the significantly larger likeli-
                                 hood as the more plausible one. The following result gives a precise state-
                                 ment.
                                    Theorem 8.3.1 (Neyman-Pearson Lemma) Consider a test of H  versus
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                                 H  stated in (8.3.1) with its rejection and acceptance regions for the null
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                                 hypothesis H  defined as follows:
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                                 or equivalently, suppose that the test function has the form






                                 where the constant k(≥ 0) is so determined that



                                 Any test satisfying (8.3.2)-(8.3.3) is a MP level α test.
                                    Proof We give a proof assuming that the X’s are continuous random vari-
                                 ables. The discrete case can be disposed off by replacing the integrals with
                                 the corresponding sums. First note that any test which satisfies (8.3.3) has
                                 size α and hence it is level a too.
                                    We already have a level a test function ψ(x) defined by (8.3.2)-(8.3.3). Let
                                 ψ*(x) be the test function of any other level α test. Suppose that Q(θ), Q*(θ)
                                 are respectively the power functions associated with the test functions ψ, ψ*.
                                 Now, let us first verify that




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                                 Suppose that x ∈ χ  is such that ψ(x) = 1 which implies L(x; θ ) - kL(x;
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                                 θ ) > 0, by the definition of ψ in (8.3.2). Also for such x, one obviously
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                                 has ψ(x) - ψ*(x) ≥ 0 since ψ*(x) ∈ (0, 1). That is, if x ∈ χ  is such that
                                 χ (x) = 1, we have verified (8.3.4). Next, suppose that x ∈ θ  is such
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                                 that χ (x) = 0 which implies L(x; θ ) - kL(x; θ ) < 0, by the definition
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