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finalold&key - ST 530 Name Final Exam All problem...

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ST 530 – May 12, 2005 Name: Final Exam All problem parts have equal weight. In budgeting your time expect that some problems will take longer than others. Remember, answers without proper justification will not receive full credit! 1. Let X 1 , . . . , X n be i.i.d. Geometric( p ) random variables. (a) Find the minimal sufficient statistics. Is it complete? (b) Find the MLE of 1 /p 2 . (Remember that if T ( X ) is the MLE of p then 1 /T 2 ( X ) is the MLE of 1 /p 2 .) (c) Find the bias of the estimator you derived in 1b. (d) Find the UMVUE of 1 /p 2 . (e) Find the asymptotic variance of the estimator you derived in 1d. Is it asymptotically efficient? (Hint: This problem requires use of both Slutsky lemma and Delta method. If you are unable to find the asymptotic variance, make sure that you at least find the CRLB for a partial credit.) (f) Consider the following prior p Beta( a, b ). Find the Bayes es- timator of 1 /p 2 . (Hint: Find the Bayes rule for loss function L ( a, p ) = ( a - 1 /p 2 ) 2 .) Solution: (a) We have a full exponential family. Therefore n i =1 X i
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