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# h7 - Stat 5102(Geyer Spring 2012 Homework Assignment 7 Due...

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Stat 5102 (Geyer) Spring 2012 Homework Assignment 7 Due Wednesday, March 21, 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. 7-1. Suppose X 1 , ... , X n are IID Cauchy( μ,σ ) and σ = 1 is known. We wish to do maximum likelihood estimation, which cannot be done in closed form, so you must use R. One needs a good estimate of the location param- eter to use as a starting point for the optimization. The location parameter is the center of symmetry and also the median. Thus the sample median is a good starting point. Data for the problem are at the URL http://www.stat.umn.edu/geyer/5102/data/prob7-1.txt (a) Find the MLE for these data. (b) Find the observed Fisher information evaluated at the MLE. (c) Find an asymptotic 95% conﬁdence interval for the parameter μ . 7-2. Suppose x 1 , ... , x n are known numbers (not random), and we ob- serve random variables Y 1 , ... , Y n that are independent but not identically distributed random variables having distributions Y i ∼ N ( α + βx i 2 ) , where

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h7 - Stat 5102(Geyer Spring 2012 Homework Assignment 7 Due...

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