Unformatted text preview: L ( X 1 ,X 2 ,...,X n  Î¾ ) denote the likelihood function of the data under parameter value Î¾ , set: Î» n = sup Î¾ âˆˆ R L ( X 1 ,X 2 ,...,X n  Î¾ ) L ( X 1 ,X 2 ,...,X n  Î¾ ) . Show that when Î¾ is the true value of Î¾ generating the data, 2 log Î» n converges to a Ï‡ 2 1 distribution, using Taylor expansions. (v) Let K ( Î¾ ,Î¾ 1 ) = E Î¾ [log ( p ( X 1 ,Î¾ ) /p ( X 1 ,Î¾ 1 ))] denote the KulbackLeibler distance between Î¾ and Î¾ 1 . Show that this is nonnegative. If Î¾ 1 6 = Î¾ is the true value of Î¾ generating the data, show that 2 log Î» n /n converges in probability to K ( Î¾ ,Î¾ 1 ). 1...
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 Fall '09
 moulib
 Probability theory, probability density function, Estimation theory, X1, Likelihoodratio test

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