sta 2006 1011_chap8 - Elementary Nonparametric Methods Dr...

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Elementary Nonparametric Methods Dr. Phillip YAM 2010/2011 Spring Semester Reference: Chapter 8: Section 1 of “Nonparametric Methods” by Hogg and Tanis.
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Section 8.1 Chi-square Goodness-Of-Fit Tests I Chi-square statistic (Karl Pearson (1900)): to test the appropriateness of different probabilistic models (as a competitor of the Kolmogorov-Smirnov test). I Y 1 follows b ( n , p 1 ) with 0 < p 1 < 1. Z = Y 1 - np 1 p np 1 (1 - p 1 ) is approximately N (0 , 1) for large n . And hence Q 1 = Z 2 is approximately χ 2 (1). I Let Y 2 = n - Y 1 and p 2 = 1 - p 1 , Q 1 = ( Y 1 - np 1 ) 2 np 1 (1 - p 1 ) = ( Y 1 - np 1 ) 2 np 1 + ( Y 1 - np 1 ) 2 n (1 - p 1 ) . ( Y 1 - np 1 ) 2 = ( n - Y 1 - n [1 - p 1 ]) 2 = ( Y 2 - np 2 ) 2 , Q 1 = ( Y 1 - np 1 ) 2 np 1 + ( Y 2 - np 2 ) 2 np 2 . I Q 1 measures the “closeness” of the observed numbers to the corresponding expected numbers.
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Section 8.1 Chi-square Goodness-Of-Fit Tests I An experiment have k mutually exclusive and exhaustive outcomes, say, A 1 , A 2 , · · · , A k . Let p i = P ( A i ) and thus k i =1 p i = 1. The experiment is repeated n independent times, Y i represent the number of times the experiment results in A i , i = 1 , 2 , · · · , k . I Joint p.d.f. of Y 1 , Y 2 , · · · , Y k - 1 : f ( y 1 , y 2 , · · · , y k - 1 ) = n ! y 1 ! y 2 ! · · · y k ! p y 1 1 p y 2 2 · · · p y k k . where y k = n - y 1 - y 2 - · · · - y k - 1 . I Q k - 1 = k X i =1 ( Y i - np i ) 2 np i . has an approximate chi-square distribution with k - 1 degrees of freedom.
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