lec7 - | | Lecture 7 We showed that ν ( S − m ( χ )) l...

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Unformatted text preview: | | Lecture 7 We showed that ν ( S − m ( χ )) l ∈ n ∀ ( ν ( S − m ( χ )) 2 ) 1 / 2 ( nI ( χ )) 1 / 2 . Next, let us compute the left hand side. We showed that ν l ∈ ( X 1 χ ) = which implies | that X ν l ∈ = ν l ∈ ( X i χ ) = n | and, therefore, ν ( S − m ( χ )) l ∈ = ν S l ∈ n − m ( χ ) ν l ∈ = ν S l ∈ . n n n Let X = ( X 1 , . . . , X n ) and denote by ( X χ ) = f ( X 1 χ ) . . . f ( X n χ ) | | | the joint p.d.f. (or likelihood) of the sample X 1 , . . . , X n We can rewrite l ∈ in terms n of this joint p.d.f. as n X ∈ ( X χ ) l ∈ = log f ( X i χ ) = log ( X χ ) = . n χ | χ | ( X | | χ ) i =1 Therefore, we can write ∈ ( X | χ ) ∈ ( X χ ) ν Sl ∈ = ν S ( X ) = S ( X ) | ( X ) dX n ( X χ ) ( X χ ) = S ( X ) ∈ ( X | χ ) dX = χ S ( X ) ( X χ ) dX = ν S ( X ) = m ∈ ( χ ) . | χ Of course, we integrate with respect to all coordinates, i.e. dX = dX 1 . . . dX n . We finally proved that m ∈ ( χ ) ∀ ( ν ( S − m ( χ )) 2 ) 1 / 2 ( nI ( χ )) 1 /...
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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.

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lec7 - | | Lecture 7 We showed that ν ( S − m ( χ )) l...

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