Understanding Newton's Multivariate Method
Explore the multivariate version of Newton's method, a generalization of the univariate version, and learn about the iterative procedure's estimation of asymptotic variance-covariance matrix. Discover how the maximum likelihood estimator becomes asymptotically normal under certain conditions.
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l n l ( ) y p i i 0 = 1 i 0 l l n ( ) y p x l 2 i i i = = = = = ( ) S 0 0 = 1 1 1 i = l l n ( ) y p x i i ip p = 1 p i
Newtons Method, Multivariate Version ( ( ) ) f x f x = 1 n x x 1 n n 1 n 1 = ( ( S )) ( ( S )) 1 1 1 n n n n
The multivariate version of Newton's method is a straight forward generalization of the univariate version. 2 2 2 l l l ... 1 = ( ( S )) ( ( S )) 0 0 0 1 0 p 1 1 1 n n n n 2 2 2 l l l ... 1 ... 0 1 ... 1 1 ... p ... ( ) is a matrix S 1 n 2 2 2 l l l 1 n ... 0 1 p p p p 2 l th The element of this matrix is ij i j
If f y is the true value of ) meets certain regularity conditions, then: 0 0 ( , , the maximum likelihood estimator of , is asymptotically normal. 1) ( , MVN I( ) 0 0 1 = ( ( S )) ( ( S )) 1 1 1 n n n n 1 2 2 2 l l l ... 0 0 0 1 0 p 2 2 2 l l l ... = ( ( S )) 1 ... 0 1 ... 1 1 ... p 1 1 n n n ... 2 2 2 l l l ... 0 1 p p p p = 1 n The iterative procedure provides an estimate of the asymptotic variance- covariance matrix
