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Unformatted text preview: Lecture 28 28.1 KolmogorovSmirnov 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 goodnessoffit test for continuous distribution but, in order to use Pearson’s theorem and chisquare 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
 DmitryPanchenko
 Statistics

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