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400    8. Tests of Hypotheses

                                    In defining a size α or level α test, what we are doing is quite simple. For
                                 every θ in the null space Θ , we look at the associated Type I error probability
                                                       0
                                 which coincides with Q(θ), and then consider the largest among all Type I
                                 error probabilities.        may be viewed as the worst possible Type I
                                 error probability. A test is called size α if     = α whereas it is
                                 called level α if          ≤ α. We may equivalently restrict our attention
                                 to a class of  test functions such that                       or
                                                  ≤ α. It should be obvious that a size a test is also a level α
                                 test.
                                    It is important to be able to compare all tests for H  versus H  such that
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                                                                                0
                                 each has some common basic property to begin with. In Chapter 7, for ex-
                                 ample, we wanted to find the best estimator among all unbiased estimators of
                                 θ. The common basic property of each estimator was its unbiasedness. Now,
                                 in defining the “best” test for H  versus H , we compare only the level a tests
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                                 among themselves.
                                    Definition 8.2.4 Consider the collection C of all level α tests for H  : θ ∈
                                                                                            0
                                 Θ  versus H  : θ ∈ Θ  where                    A test belonging to C
                                   0
                                                   1
                                           1
                                 with its power function Q(θ) is called the best or the uniformly most powerful
                                 (UMP) level α test if and only if Q (θ) ≥ Q* (θ) for all θ ∈ Θ where Q* (θ)

                                 is the power function of any other test belonging to C. If the alternative
                                 hypothesis is simple, that is if Θ  is a singleton set, then the best test is called
                                                            1
                                 the most powerful (MP) level a test.
                                    Among all level a tests our goal is to find the one whose power hits the
                                 maximum at each point θ ∈ Θ . Such a test would be UMP level α. In the
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                                 simple null versus simple alternative case first, Neyman and Pearson (1933a,b)
                                 gave an explicit method to determine the MP level α test. This approach is
                                 described in the next section. We end this section with an example.
                                    Example 8.2.4 (Example 8.2.1 Continued) Consider a population with the
                                 pdf N(θ, 1) where θ ∈ ℜ is unknown. Again an experimenter postulates two
                                 hypotheses, H  : θ = 5.5 and H  : θ = 8. A random sample X = (X , ..., X ) is
                                                                                         1
                                                                                              9
                                                           1
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                                 collected. Let us look at the following tests:




                                 Let us write Q (θ) for the power function of Test #i, i = 1, 2, 3. Using
                                              i
                                 MAPLE, we verified that Q (5.5) = Q (5.5) = Q (5.5) = .049995. In other
                                                          1
                                                                           3
                                                                  2
                                 words, these are all level a tests where α = .049995. In the Figure 8.2.1,
                                 we have plotted these three power functions. It is clear from this plot that
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