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### hw5_stat210a

Course: STAT 210a, Fall 2006
School: Berkeley
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Word Count: 470

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Berkeley UC Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 5 Fall 2006 Issued: Thursday, September 28, 2006 Due: Thursday, October 5, 2006 A useful definition for this problem set: Definition: An equalizer procedure is a rule with constant risk (i.e., R(, ) = c for all ). Problem 5.1 Let = (0, +) and A = [0, ), and suppose that X Poi(). Consider the loss function L(,...

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Berkeley UC Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 5 Fall 2006 Issued: Thursday, September 28, 2006 Due: Thursday, October 5, 2006 A useful definition for this problem set: Definition: An equalizer procedure is a rule with constant risk (i.e., R(, ) = c for all ). Problem 5.1 Let = (0, +) and A = [0, ), and suppose that X Poi(). Consider the loss function L(, a) = ( - a)2 /. (a) Show that the estimator (X) = X is an equalizer rule. (b) Show that is generalized Bayes with respect to the improper prior that is uniform on (0, ). (c) Find Bayes estimators with respect to the family of Gamma(a, b) priors. Problem 5.2 For (0, 1), let X Bin(n, ), and consider the weighted quadratic loss function L(, a) = ( - a)2 . (1 - ) (Note that this loss function penalizes more severely for extreme values of near 0 or 1.) (a) Show that for this loss function, (X) = X/n is a Bayes estimator with respect to the uniform distribution on [0, 1]. (b) Show that for this loss function, is a minimax estimate of , with constant risk 1/n. Problem 5.3 Consider the Bayesian model in which, conditioned on = , we have X Bin(n, ), and we place a Beta(a, b) prior on . Recall that in a previous homework, you showed that for the Bayes estimator a,b under quadratic loss, the associated risk function takes the form R(, a,b ) = 2 (a + b)2 - n + [n - + 2a(a b)] + a2 . (n + a + b)2 Use this form of the risk function to find a minimax estimator of . (Hint: Think of choices of a and b that lead to equalizer rules.) 1 Problem 5.4 Consider the decision-theoretic problem with = [0, 1] action space A = [0, 1], and loss function L(, a) = (1 - ) a + (1 - a). Conditioned on = , let X have any distribution P . Show that the rule (X) = 1/2 is a minimax rule. (This example illustrates that minimax rules need not be good rules.) Problem 5.5 This problem addresses the issue of implementing Bayes estimators for exponential family models. Suppose that we have a (conditional) exponential family model d p(x | ) = h(x) exp i=1 i Ti (x) - A() , where x = (x1 , . . . , xn ) and has density (). (a) Define the marginal density m(x) = Show that for j = 1, . . . , n, we have d p(x | )()d induced by this Bayesian model. E i=1 i Ti (x) | x xj = log m(x) - log h(x). xj xj (Assume here that all relevant quantities are suitably differentiable.) d (b) Suppose that X = (X1 , . . . , Xn ) has density p(x; ) = h(x) exp i=1 i xi - A() . Use part (a) to conclude that the Bayes estimator of j under quadratic loss is given by (x) = log m(x) - log h(x). xj xj (c) Apply your result from (b) to derive the Bayes estimator under quadratic loss for the normal-normal model with Xi | N (, 2 ), i = 1, . . . n i.i.d., and N (, 2 ). 2
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Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 5 Fall 2006 Issued: Thursday, September 14, 2006 Due: Thursday, September 21, 2006Graded exercisesProblem 5.1 a) We want to show that R(, ) = E ( (X-) )
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 6 Fall 2006 Issued: Thursday, October 5, 2006 Due: Thursday, October 12, 2006Problem 6.1 In the inverse binomial sampling procedure, N is a random variabl
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 6 Fall 2006 Issued: Thursday, October 5, 2006 Problem 6.1 a) Using the Rao-Blackwell theorem and considering the quadratic error loss function L(, ) = ( -
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 7 Fall 2006 Issued: Thursday, October 19, 2006 Due: Thursday, October 26, 2006Problem 7.1 p d d Show that if Xn X &gt; 0 and Xn /Yn 1, then Yn X. Problem 7.2
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Solutions - Problem Set 7 Fall 2006 Issued: Thursday, September 14, 2006 Due: Thursday, September 21, 2006Graded exercisesProblem 7.1 p Y Let Zn = Xn , we have that
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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Solutions - Problem Set 8 Fall 2006 Issued: Thursday, November 2, 2006 Due: Thursday, November 9, 2006Graded exercisesProblem 8.1 (a) From the definitions given, we
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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 9 Fall 2006 Issued: Thursday, November 2, 2006 Due: Thursday, November 9, 2006Some useful notation: Let denote the CDF of a standard normal variate, and l
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Solutions - Problem Set 9 Fall 2006 Issued: Thursday, November 2, 2006 Due: Thursday, November 9, 2006Graded exercisesProblem 9.1 (a) First, notice that ga () = P(X1
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 10 Fall 2006 Issued: Thursday, November 9, 2006 Due: Thursday, November 16, 2006Problem 10.1 (A non-parametric hypothesis test) A set of i.i.d. samples Y1
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Solutions - Problem Set 10 Fall 2006 Issued: Thursday, November 9, 2006 Due: Thursday, November 16, 2006Graded exercisesProblem 10.1 a) For each i, Zi = I(Yi &gt; 0 ) f
Berkeley - STAT - 210a
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Berkeley - STAT - 210a
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Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Midterm Examination-Solutions Fall 2006 Problem 1.1 [18 points total] Suppose that Xi , i = 1, . . . , n are i.i.d. samples from the uniform Uni[0, ] distribution. (a)
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 1 Fall 2006 Issued: Thursday, August 31, 2006 Due: Thursday, September 7, 2006Problem 1.1 p Given a sequence of random variables such that Yn , give one e
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 2 Fall 2006 Issued: Thursday, September 7, 2006 Due: Thursday, September 14, 2006Graded problemsProblem 2.1 Suppose that Xi , i = 1, . . . , n are i.i.d.
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 3 Fall 2006 Issued: Thursday, September 14, 2006 Due: Thursday, September 21, 2006GradedProblem 3.1 The inverse Gaussian distribution IG(, ) has density
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 4 Fall 2006 Issued: Thursday, September 21, 2006 Note: For this problem set, &quot;Norway&quot;. Problem 4.1 By Taylor series expansion, we have the identity - log(1
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 8 Fall 2006 Issued: Thursday, October 26, 2006 Due: Thursday, November 2, 2006Some useful notation: The pth quantile of a continuous random variable with
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