Homework6-610 - L ( X 1 ,X 2 ,...,X n | ) denote the...

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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 | ) denote the likelihood function of the data under parameter value , set: n = sup R L ( X 1 ,X 2 ,...,X n | ) L ( X 1 ,X 2 ,...,X n | ) . Show that when is the true value of generating the data, 2 log n converges to a 2 1 distribution, using Taylor expansions. (v) Let K ( , 1 ) = E [log ( p ( X 1 , ) /p ( X 1 , 1 ))] denote the Kulback-Leibler distance between and 1 . Show that this is non-negative. If 1 6 = is the true value of generating the data, show that 2 log n /n converges in probability to K ( , 1 ). 1...
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