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AMDA-2008 Session 6

# AMDA-2008 Session 6 - Parametric Inference II...

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Parametric Inference – II

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Kullback-Leibler distance )) f(x; ), D(f(x; for ) , D( write shall we , , For 0. f) D(f, and 0 g) D(f, shown that be can It g(x) f(x) f(x)ln g) D(f, be to defined is g and f between distance Leibler - Kullback the s pdf' are g and f If ψ θ ψ θ ψ θ Θ = = dx
Identifiability ons. distributi different to correspond parameter the of values different that means This 0 ) , D( that implies if le identifiab be to said is model The ψ θ ψ θ

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Consistency of MLE - = = - = → - = = θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ when maximized is which ) , ( ) ; ( ) ; ( ln E Now, ) ; ( ) ; ( ln E ) ( M By WLLN, ). (w.r.t constant a is ) ( and )) ( ) ( ( ) ( M since ) ( M maximizing to equivalent is ) ( Maximizing ) ; ( ) ; ( ln 1 ) ( M Define . of value true the denote Let P n * 1 n n n 1 n D X f X f X f X f l l l n l X f X f n i i i i n n n n i i i
Consistency of MLE

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Equivariance of the MLE . of MLE the is ) ˆ ( ˆ Then . of MLE the be ˆ Let . of function a be ) g( Let n n τ θ τ θ θ θ θ τ n g = =
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