AMDA-2008-Session 10 - Goodness of Fit Tests Multinomial...

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Goodness of Fit Tests
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Multinomial Distribution integral. are 0 ,..., ) ... (x ... ! !... ! ) ,..., P(X if ) 1 ... (p ,..., p n, parameters with on distributi l multinomia a follow ) ,..., (X X say We 1 1 1 1 1 1 1 1 1 1 = + + = = = = + + = k k x k x k k k k k k x x n x p p x x n x X x p p X k
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Estimation of Parameters n x p n p x n p p x n p l p x n p x p l p x n p x C p p l x x n p p p x C p p l i i k i i k i i k i i i k i i i k i i k i i i i i k i i k i i k i i i k k k i k i i k = = - - = - = - = = - - - = - - + + = - - - = - - - = + = - = - = - = - = - = - = - = - = - = - - - = ˆ ) 1 ( ) ( get we 1 - k 1,..., i for equations above the Adding ) 1 ( ) ( x get we 0 Putting 1) - k 1,..., (i ) 1 ( ) ( ) 1 ln( ) ( ln ) ,..., ( get we ... x and ... 1 p Putting ln ) ,..., ( 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 k 1 1 k 1 1
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Important properties j i j i i i i k k p np X X Cov p np np p p n Mult X - = - = = = ) , ( ) 1 ( ) SD(X ) E(X ) p Bin(n, is X of on distributi marginal the ) ,..., , ( ~ ) ,..., (X X Suppose i i i i 1 1
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χ 2 -Goodness of Fit test . as d distribute is T , H under n, large for Then . ) (O T and E Define . H under ) ,..., , ( ~ ) ,..., (O Note . x value taken the has that sample in the ' X of number observed the be O Let : H e alternativ the versus F F : H hypothesis null test the want to .We ,..., x values the taking F on distributi discrete some from sample random a be X ,..., X Let . (known) ,..., p ies probabilit with ,..., x values the es which tak F on distributi discrete a
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