Homework6-610 - L X 1,X 2,X n | ξ denote the likelihood...

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Homework 6 Announcement: The Homework carries a total of 50 points. Let X 1 ,X 2 ,...,X n ,... be i.i.d. observations from a one parameter exponential family model p ( x,ξ ) = exp( ξ T ( x ) - A ( ξ )) h ( x ). Let ξ 0 be the true underlying value of the parameter. (i) Find the mean and variance of T ( X 1 ). Suggest a method of moments estimator for ξ 0 . (ii) Find the MLE of ξ 0 . Argue that ”the MLE” makes sense, i.e. the MLE is unique. What is the connection between the MOM and the MLE in this model? (iii) Find the limit distribution of n ( ˆ ξ MLE - ξ 0 ). What is the connection between the variance of the limit distribution and the information bound for estimating ξ 0 in this model? (iv) Consider the likelihood ratio test for testing H 0 : ξ = ξ 0 in this model, against H 1 : ξ 6 = ξ 0 . Thus, letting
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Unformatted text preview: L ( X 1 ,X 2 ,...,X n | ξ ) denote the likelihood function of the data under parameter value ξ , set: λ n = sup ξ ∈ R L ( X 1 ,X 2 ,...,X n | ξ ) L ( X 1 ,X 2 ,...,X n | ξ ) . Show that when ξ is the true value of ξ generating the data, 2 log λ n converges to a χ 2 1 distribution, using Taylor expansions. (v) Let K ( ξ ,ξ 1 ) = E ξ [log ( p ( X 1 ,ξ ) /p ( X 1 ,ξ 1 ))] denote the Kulback-Leibler distance between ξ and ξ 1 . Show that this is non-negative. If ξ 1 6 = ξ is the true value of ξ generating the data, show that 2 log λ n /n converges in probability to K ( ξ ,ξ 1 ). 1...
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