stat210a_fall07_hw8

# stat210a_fall07_hw8 - UC Berkeley Department of Statistics...

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Unformatted text preview: UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 8 Fall 2007 Issued: Thursday, October 25 Due: Thursday, November 1 Reading: Keener: Chapter 10, 11, § 21.6, B & D: § 3.5, § 4.4 Problem 8.1 Find variance-stabilizing transformations for the following statistics and estimation problems: (a) The sample mean ¯ X n for i.i.d. samples from Ber( θ ). (Here we want the transformed statistic to have asymptotic variance independent of θ .) (b) The sample correlation coefficient r = P n i =1 X i Y i √ P n i =1 X 2 i √ P n i =1 Y 2 i for i.i.d. samples from the zero-mean bivariate normal with σ 2 X = σ 2 Y = 1 and correlation coefficient ρ ∈ ( − 1 , 1). (Here we want the transformed statistic to have asymptotic variance independent of ρ .) Problem 8.2 Let X 1 ,...,X n be i.i.d. with E [ X i ] = θ , var( X i ) = 1, and E [( X i − θ ) 4 ] = μ 4 , and consider the unbiased estimators of θ 2 defined by δ 1 ( X ) = 1 n n summationdisplay i =1 X 2 i − 1 , and δ 2 (...
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stat210a_fall07_hw8 - UC Berkeley Department of Statistics...

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