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Unformatted text preview: Stat 5102 (Geyer) Spring 2012 Homework Assignment 4 Due Wednesday, February 15, 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. 4-1. Calculate the ARE of the sample mean X n versus the sample median e X n as an estimator of the center of symmetry for (a) The Laplace location-scale family having density given in the brand name distributions handout. (b) The t ( ) location-scale family, with densities given by f ,, ( x ) = 1 f x- where f is the t ( ) density given in the brand name distributions hand- out. (Be careful to say things that make sense even considering that the t ( ) distribution does not have moments of all orders. Also is not the standard deviation.) (c) The family of distributions called Tri( , ) (for triangle) with densities f , ( x ) = 1 1- | x- | , | x- | < shown below H H H H H H H H H H H - + 1 / The parameter can be any real number, must be positive. 4-2. Let X 1 , X 2 , ... , X n be an IID sample having the N ( , 2 ) distribu- tion, where and 2 are unknown parameters, and let S 2 n denote the sample variance (defined as usual with n- 1 in the denominator). Suppose n = 5 and S 2 n = 53 . 3. Give an exact (not asymptotic) 95% confidence interval for 2 ....
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- Spring '03