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Unformatted text preview: 18.466, Dudley March 11, 2003 CHAPTER 1. DECISION THEORY AND TESTING SIMPLE HYPOTHESES 1.1 Deciding between two simple hypotheses: the Neyman-Pearson Lemma . Probability theory is reviewed in Appendix D. Suppose an experiment has a set X of possible outcomes. The outcome has some probability distribution µ defined on X . In statistics, we typically don’t know what µ is, but we have hypotheses about what it may be. After making observations we’ll try to make a decision between or among the hypotheses. In general there could be infinitely many possibilities for µ , but to begin with we’re going to look at the case where there are just two possibilities, µ = P or µ = Q , and we need to decide which it is. For example, a point x in X could give the outcome of a test for a certain disease, where P is the distribution of x for those who don’t have the disease and Q is the distribution for those who do. Often, we have n observations independent with distribution µ . Then X can be replaced by the set X n of all ordered n-tuples ( x 1 , . . . , x n ) of points of X , and µ by the Cartesian product measure µ × ··· × µ of n copies of µ . In this way, the case of n observations x 1 , . . . , x n reduces to that of one “observation” ( x 1 , . . . , x n ). The probability measures P and Q are each defined on some σ-algebra B of subsets of X , such as the Borel sets in case X is the real line R or a Euclidean space. A test of the hypothesis that µ = P will be given by a measurable set A , in other words a set A in B . If we observe x in A , then we will reject the hypothesis that µ = P in favor of the alternative hypothesis that µ = Q . Then P ( A ) is called the size of the test A (at P ). The size is the probability that we’ll make the error of rejecting P when it’s true, i.e. when µ = P , sometimes called a Type I error . On the other hand, Q ( A ) is called the power of the test A against the alternative Q . The power is the probability that when Q is true, the test correctly rejects P and prefers Q . The complementary probability 1 − Q ( A ) is sometimes called the probability of a Type II error . Given P and Q , for the test A to be as effective as possible, we’d like the size to be small and the power to be large. In the rest of this section, it will be shown how the choice of A can be made optimally. Example 1.1.1. Let X = R and let P and Q be normal measures, both with variance . 04, P = N (0 , . 04) and Q = N (1 , . 04). Larger values of x tend to favor Q , so it seems reasonable to take A as a half-line [ c, ∞ ) for some c . At x = 1 / 2 , the densities of P and Q are equal. For x < 1 / 2 , P has larger density. For x > 1 / 2, Q does. So if we have no reason in advance to prefer one of P and Q , we might take c = 1 / 2. Then the probabilities of the two types of errors are each about . 0062 (from tables of the normal distribution)....
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This note was uploaded on 09/19/2011 for the course MATH 111 taught by Professor Jj during the Spring '09 term at AIU Online.
- Spring '09