# HW5 - HW#5 Due Wednesday by 5:00 pm(slip under instructor's...

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1 HW #5 Due Wednesday, 11/2/16 by 5:00 pm (slip under instructor's door) Read Notes_4.pdf and the rest of Sections 6.2 and 6.4 (for the C-R lower bound), Sections 7.2, 7.3, and 7.7 (for sufficiency and the R-B theorem), and Sections 8.1 and 8.2 (for most powerful and uniformly most powerful tests). Make sure to read all the examples in the textbook. 1) Consider a random sample { X i : i = 1, 2, . . ., n } from a "shifted" exponential distribution with pdf ( ) < < = : 0 : 1 1 1 2 1 2 2 1 θ θ θ θ θ θ θ x x x exp , ; x f where −∞ < θ 1 < , and 0 < θ 2 < . a) Explain how we can view θ 1 as a "location" parameter and θ 2 as a "scale" parameter. b) Find a minimal sufficient statistic for θ = [ θ 1 , θ 2 ]. 2) Consider a random sample { X i : i = 1, 2, . . ., n } ~ NID ( µ , σ 2 ). a) Is the MLE of θ = [ µ , σ 2 ] ' efficient? Asymptotically efficient? Hint: The variance of a chi-square random variable is twice its degrees-of-freedom. b) Is the MLE of µ efficient? Asymptotically efficient?

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• Fall '16
• Variance, Probability theory, exponential family, rao-blackwell theorem

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