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# h4 - Stat 5102(Geyer Spring 2010 Homework Assignment 4 Due...

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Stat 5102 (Geyer) Spring 2010 Homework Assignment 4 Due Wednesday, February 17, 2010 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 ( ν ) location-scale family, with densities given by f ν,μ,σ ( x ) = 1 σ f ν ± x - μ σ ² where f ν is the t ( ν ) density given in the brand name distributions hand- out. (Be careful to say things that make sense even considering that the t ( ν ) distribution does not have moments of all orders. Also σ is not the standard deviation.) (c) The family of distributions called Tri( μ,λ ) (for triangle) with densities f μ,λ ( x ) = 1 λ ± 1 - | x - μ | λ ² , | x - μ | < λ shown below ± ± ± ± ± ± ± ± ± ± ± H H H H H H H H H H H μ μ - λ μ + λ 0 1 The parameter μ can be any real number, λ must be positive. 4-2. Let X 1 , X 2 , ... , X n be an IID sample having the N ( μ,σ 2 ) distribu- tion, where μ and σ 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 σ 2 .

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h4 - Stat 5102(Geyer Spring 2010 Homework Assignment 4 Due...

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