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Unformatted text preview: Lecture 28 28.1 Kolmogorov-Smirnov test. Suppose that we have an i.i.d. sample X 1 , . . . , X n with some unknown distribution and we would like to test the hypothesis that is equal to a particular distribution , i.e. decide between the following hypoheses: H 1 : = H 2 : otherwise We considered this problem before when we talked about goodness-of-fit test for continuous distribution but, in order to use Pearson’s theorem and chi-square test, we discretized the distribution and considered a weaker derivative hypothesis. We will now consider a different test due to Kolmogorov and Smirnov that avoids this discretization and in a sense is more consistent. Let us denote by F ( x ) = ( X 1 ≤ x ) a cumulative distribution function and consider what is called an empirical distribution function: F n ( x ) = n ( X ≤ x ) = 1 n n X i =1 I ( X i ≤ x ) that is simply the proportion of the sample points below level x. For any fixed point x ∈ the law of large numbers gives that F n ( x ) = 1 n n X i =1 I ( X i ≤ x ) → I ( X 1 ≤ x ) = ( X 1 ≤ x ) = F ( x ) , i.e. the proportion of the sample in the set (-∞ , x ] approximates the probability of this set. It is easy to show from here that this approximation holds uniformly over all x ∈ : sup x ∈ | F n ( x )- F ( x ) | → 110 LECTURE 28. 111 1 y x X Y F(X) = Y Figure 28.1: C.d.f. and empirical d.f....
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This note was uploaded on 10/11/2009 for the course STATISTICS 18.443 taught by Professor Dmitrypanchenko during the Spring '09 term at MIT.
- Spring '09