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stat210a_fall07_hw7

# stat210a_fall07_hw7 - UC Berkeley Department of Statistics...

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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 7 Fall 2007 Issued: Thursday, October 25 Due: Thursday, October 18 Reading: Keener: Chapter 10, 11. B & D: Chapter 5. Problem 7.1 Suppose that X 1 , . . . , X n are i.i.d. N (0 , σ 2 ). (a) Show that δ 0 ( X ) = C n n i =1 | X i | is a consistent estimator of σ if and only if C = p π/ 2. (b) Show that the MLE of σ is given by δ MLE ( X ) = q 1 n n i =1 X 2 i . Determine the asymp- totic relative efficiency of δ 0 with c = p π/ 2 compared to the MLE δ MLE . Problem 7.2 Suppose that X 1 , . . . X n are i.i.d from the normal location model N ( θ, 1), and that we wish to estimate the critical or cutoff value g a ( θ ) = P [ X 1 a ], where a R is some fixed number. (a) Let Φ denote the CDF of the standard normal distribution. Show that the estimator δ n ( X ) = Φ r n n - 1 ( a - ¯ X n ) is an unbiased estimator of g a ( θ ). Prove that n ( δ n - g a ( θ )) d N (0 , φ 2 ( a - θ )), where φ is the PDF of the standard normal.

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stat210a_fall07_hw7 - UC Berkeley Department of Statistics...

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