411_hw07 - below , argue that [ m ( θ ) is the MVUE of m (...

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Stat 411 – Homework 07 Due: Wednesday 03/07 Undergraduates may solve the “Graduate only” problem(s) for possible extra credit. 1. Let X 1 ,...,X n be iid N ( θ, 1). Find the MVUE of η = θ 2 . 2. Problem 7.5.3 on page 392. 3. Problem 7.5.6 on page 393. 4. Problem 7.5.13 on page 394. 5. (Graduate only) Let X 1 ,...,X n be iid from a distribution with PDF/PMF f θ ( x ) in the (regular) exponential family, i.e., f θ ( x ) = exp ± p ( θ ) K ( x ) + S ( x ) + q ( θ ) ² . Define m ( θ ) = E θ [ K ( X 1 )] and the estimator [ m ( θ ) = 1 n n i =1 K ( X i ). (a) Without using part (b)
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Unformatted text preview: below , argue that [ m ( θ ) is the MVUE of m ( θ ). (b) Show that the variance of [ m ( θ ) is equal to the Cramer–Rao lower bound. (Hints: You’ll need to calculate the Fisher information I ( θ ) for f θ ( x ) define above. Also, remember that m ( θ ) is a function of the parameter θ , so the Cramer–Rao lower bound is not simply [ nI ( θ )]-1 . Finally, the results in The-orem 7.5.1 should be helpful.) 1...
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This note was uploaded on 03/12/2012 for the course STAT 411 taught by Professor Staff during the Spring '08 term at Ill. Chicago.

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