notes18 - Statistics 512 Notes 18 Multiparameter maximum...

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Statistics 512 Notes 18: Multiparameter maximum likelihood estimation We consider 1 , , n X X K iid with pdf ( ; ), f x  where 1 ( , , ) p K is p-dimensional. As before, 1 1 1 1 ( ) ( ; , , ) ( ) log ( ) log ( ; , , ) n i p i n i p i L f x l L f x K K The maximum likelihood estimate is ˆ arg max ( ) arg max ( ) MLE L l   We can find critical points of the likelihood function by solving the vector equation 1 1 1 2 1 ( , , ) 0 ( , , ) 0 ( , , ) 0 p p p p l l l K K M K We need to then verify that the critical point is a global maximum. Example 1: Normal distribution
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1 , , n X X K iid 2 ( , ) N   2 2 1 1 ( ) 1 1 ( , , ; , ) exp 2 2 n i n i x f x x   K 2 2 1 1 ( , ) log log 2 ( ) 2 2 n i i n l n X     The partials with respect to and are 2 1 3 2 1 1 ( ) ( ) n i i n i i l X l n X   Setting the first partial equal to zero and solving for the mle, we obtain ˆ MLE X Setting the second partial equal to zero and substituting the mle for , we find that the mle for is 2 1 1 ˆ ( ) n MLE i i X X n . To verify that this critical point is a maximum, we need to check the following second derivative conditions: (1) The two second-order partial derivatives are negative: 2 2 ˆ ˆ , 0 MLE MLE l and 2 2 ˆ ˆ , 0 MLE MLE l
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(2) The Jacobian of the second-order partial derivatives is positive, 2 2 2 2 2 2 ˆ ˆ , 0 MLE MLE l l l l        
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