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# h3 - Stat 5102(Geyer Spring 2012 Homework Assignment 3 Due...

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Stat 5102 (Geyer) Spring 2012 Homework Assignment 3 Due Wednesday, February 8, 2012 Solve each problem. Explain your reasoning. No credit for answers with no explanation. If the problem is a proof, then you need words as well as formulas. Explain why your formulas follow one from another. 3-1. Show that the family of Gam( α, λ ) distributions with α known and λ unknown, so the parameter space is { λ R : λ > 0 } is a scale family. 3-2. Suppose S 2 n is the sample variance calculated from an IID normal random sample of size n . (a) Calculate the bias of S n as an estimator of the population standard deviation σ . (b) Find the constant a such that aS n has the smallest mean square error as an estimator of σ . 3-3. Suppose U and V are statistics that are independent random variables and both are unbiased estimators of a parameter θ . Write var( U ) = σ 2 U and var( V ) = σ 2 V , and define another statistic T = aU + (1 - a ) V where a is an arbitrary but known constant.

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