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### HW04

Course: STAT 210A, Fall 2009
School: Berkeley
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Word Count: 371

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Berkeley, UC Department of Statistics STAT 210A: Introduction to Mathematical Statistics HW#4 Fall, 2009 Due: Tuesday, September 29, 2009 4.1. Let X 1 ,.., X n be independent with X i ~ N (ti ,1) , where t1 ,..., tn are a sequence of known constants (not all zero). (a) Show that the least squares estimator, = ti X i / ti2 is complete sufficient for the family of joint distributions. (b) Use Basus theorem to...

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Berkeley, UC Department of Statistics STAT 210A: Introduction to Mathematical Statistics HW#4 Fall, 2009 Due: Tuesday, September 29, 2009 4.1. Let X 1 ,.., X n be independent with X i ~ N (ti ,1) , where t1 ,..., tn are a sequence of known constants (not all zero). (a) Show that the least squares estimator, = ti X i / ti2 is complete sufficient for the family of joint distributions. (b) Use Basus theorem to show that and (X i ti ) are independent. 2 4.2. The inverse Gaussian distribution IG ( , ) has density function 1 exp( ) x 3 / 2 exp ( + x ) , 2 2x x > 0, > 0, > 0. (a) Show that this density constitutes an exponential family. n 1 1 1n (b) Show that the statistic T = ( X = X i , S = ( ) ) is complete and sufficient. n i =1 X i =1 X i 4.3. Determine the natural parameter space of the associated exponential family of dimension one with = , T ( x ) = x and (a) h(x) = exp(- x 2 ). (b) h(x) = exp(-|x|). (c) h(x) = exp(-|x|)/(1+ x 2 ). 4.4. Let Z be distributed according to N ( , 2 ) with and 2 unknown. For a given and 2 + , define G , ( x ) = P( Z x | Z > 0) . 2 (a) Prove G that is a cdf for any given , 2 . (b) Prove that the distribution G belongs to the exponential family. (c) Find the moment generating function of the distribution G and compute the mean and variance of G. 4.5. (a) Given a k-dimensional random vector x, prove that the distribution on {1, 2, 3, } given by p( x; ) = k 1{max x j } is not an exponential family. j (b) Is the uniform distribution on [0, ] an exponential family? Why or why not? 4.6. Suppose that X has density in exponential family form p( x, ) exp ( d i =1 i Ti ( x ) A( ) h( x ), ) where each Ti is differentiable. (a) Prove the following form of Stein's identity. If X has support on the real line and g is any differentiable function such that E | g '( X ) | < + , then h '( X ) d E + i Ti ' ( X ) g ( X ) = E[ g '( X )]. h ( X ) i =1 2 (b) Specializing to the N ( , ) case, use part (a) to show that Cov ( g ( X ), X ) = 2 E [ g '( X )] . (c) Use part (b) to compute the third and fourth moments of the N ( , 2 ) distribution. The following problems will not be graded Bickel & Doksum, 1.6.3, 1.6.5, 1.6.10 (pages 87 & 88)
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Berkeley - STAT - 210A
UC Berkeley, Department of Statistics STAT 210A: Introduction to Mathematical Statistics HW#5 4.1. By Taylor series expansion, we have the identity log(1 ) = all (0,1) . From this fact, the quantityFall, 2009Due: Tuesday, October 06, 2009 x =1xx, wh
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STAT210A HW06Due: Tuesday, October 13, 2009A useful denition for this problem set: Denition: An equalizer procedure is a rule with constant risk (i.e., R(, ) = c for all ). 6.1. Consider the Bayesian model in which has distribution , and conditioned on
Berkeley - STAT - 210A
STAT210A HW07Due: Tuesday, October 20, 20097.1.In the inverse binomial sampling procedure, N is a random variable representing the number of trials required to observe x successes in a total of N + x Bernoulli trials (with parameter ). (a) Show that the
Berkeley - STAT - 210A
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 1 - Solutions Fall 2009 Issued: Tuesday, September 1, 2009 Problem 1.1 1. Yn = 0, n, with probability 1 1 with probability n .1 nDue: Tuesday, September
Berkeley - STAT - 210A
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 2 - Solutions Fall 2009 Issued: Tuesday, September 8, 2009 Problem 2.1 From the distribution of X, we have:nDue: Tuesday, September 15, 2009P (X = x) =
Berkeley - STAT - 210A
UC Berkeley Department of StatisticsSTAT 210A: Introduction to Mathematical Statistics Problem Set 3 - Solutions Fall 2009 Issued: Tuesday, September 15, 2009 Problem 3.1 Let X N (, 1). From, 1 1 2 p (x) = exp ( x2 + x ), 2 2 2 T (X ) = X is sucient by t
Berkeley - STAT - 210A
UC Berkeley Department of StatisticsSTAT 210A: Introduction to Mathematical Statistics Problem Set 4 - Solutions Fall 2008 Issued: Tuesday, September 22, 2009 Problem 4.1 (a) The joint distribution is,nDue: Tuesday, September 29, 2009p(x; ) =i=1 nP
Berkeley - STAT - 210A
UC Berkeley Department of StatisticsSTAT 210A: Introduction to Mathematical Statistics Problem Set 5 - Solutions Fall 2008 Issued: Tuesday, September 29, 2009 Problem 5.1 First, notice that we can rewrite: p(x; ) = exp [x log log ( log(1 )] so p belongs
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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 1- Solutions Fall 2007 Issued: Thursday, August 30 Due: Thursday, September 6 Problem 1.1Pn (a) We have that E (X )2 = EPni=1 n2 2 i2 i=1 (Xi ) 2 n .
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 2- Solutions Fall 2007 Issued: Thursday, September 6 Due: Thursday, September 13 Problem 2.1 (a) Let fi (x) be the density of Xi , X = (X1 , . . . , Xn ) a
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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 3- Solutions Fall 2007 Issued: Thursday, September 13 Due: Thursday, September 20 Problem 3.1 (a) Let FX be distribution of X . Then, FX (x) = 1 ex for all
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 4- Solutions Fall 2007 Issued: Thursday, September 20 Due: Thursday, September 27 Problem 4.1 1c (a) P(X x) = F (x) = 1 1(x &gt; 1). x f (x) = cxc1 1(x &gt; 1) =
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 5- Solutions Fall 2007 Issued: Thursday, September 27 Due: Thursday, October 4 Problem 5.1 (a) For the Gamma density, we have: p(x|) exp log(x) px + log p
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 6- Solutions Fall 2007 Issued: Thursday, October 4 Due: Thursday, October 11 Problem 6.1 Note: You can NOT apply the Theorem 11.1 with improper prior. Plea
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 7- Solutions Fall 2007 Issued: Thursday, October 18 Due: Thursday, October 25 Problem 7.1 (a)E(|Xi |) = 2 =0xx2 1 e 22 dx = 2 0t e 2 dt 2t=x2 22 C
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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 8- Solutions Fall 2007 Issued: Thursday, October 25 Due: Thursday, December 1 Problem 8.1 (a) For a Bernoulli random variable, we have: E X = var X = (1 )
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 9- Solutions Fall 2007 Issued: Thursday, December 1 Due: Thursday, December 8 Problem 9.1 We want to prove that, for all , P ( S (X ) 1 . To prove that, we
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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 10- Solutions Fall 2007 Issued: Thursday, December 8 Due: Thursday, December 15 Problem 10.1 a) The action space is given by A = cfw_0, 1. For = 0 , the lo
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 11- Solutions Fall 2007 Issued: Thursday, December 19 Due: Thursday, December 29 Problem 11.1 (a) P0 1 nn i=1Xi 0 = P0 etPni=1Xi etn(0 +) , t &gt; 0Xi
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 1 Fall 2007 Issued: Thursday, August 30d p m.s.Due: Thursday, September 6Notation: The symbols cfw_, , denote convergence in distribution, probability
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 2 Fall 2007 Issued: Thursday, September 6 Due: Thursday, September 13Problem 2.1 Suppose that (X1 , X2 , . . . , Xn ) is multivariate normal with unknown
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 3 Fall 2007 Issued: Thursday, September 13 Due: Thursday, September 20Problem 3.1 Consider an exponentially distributed variate with density fX (t) : = ex
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 4 Fall 2007 Issued: Thursday, September 20 Due: Thursday, September 27Problem 4.1 A random variable X has the Pareto distribution P (c, k ) if its CDF tak
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 5 Fall 2007 Issued: Thursday, September 27 Due: Thursday, October 4Reading: For this problem set: Chapter 9 of Keener (Bayesian methods); 3.1, 3.2 of Bick
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 6 Fall 2007 Issued: Thursday, October 4 Due: Thursday, October 11Reading: Keener, Chapter 11. Bickel and Doksum; 2.12.3. 3.13.3 Problem 6.1 1 Consider n i
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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 7 Fall 2007 Issued: Thursday, October 25 Reading: Keener: Chapter 10, 11. B &amp; D: Chapter 5. Problem 7.1 Suppose that X1 , . . . , Xn are i.i.d. N (0, 2 ).
Berkeley - STAT - 210a
UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 8 Fall 2007 Issued: Thursday, October 25 Reading: Keener: Chapter 10, 11, 21.6, B &amp; D: 3.5, 4.4 Problem 8.1 Find variance-stabilizing transformations for t
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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 9 Fall 2007 Issued: Thursday, November 1 Reading: Keener: Chapter 14; B &amp; D: 3.5, 4.4 Problem 9.1 Suppose that for each 0 , the set A(0 ) is the acceptance
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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 10 Fall 2007 Issued: Thursday, November 8 Reading: Bickel &amp; Doksum: 6.3, 6.4; Keener: Chapter 18 Problem 10.1 Suppose that we use a prior = [0 (1 0 )] for
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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 1- Solutions Fall 2006 Issued: Thursday, August 31, 2006 Problem 1.1 Solution to 1. Let: Yn = 0, n, with probability 1 1 with probability n 1 nDue: Thurs
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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, 2006Graded exercisesProblem 3.1 (a) To prove that, we massage the density e
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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-) )
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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
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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(, ) = ( -
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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
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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
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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
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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
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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 11 Fall 2006 Issued: Thursday, November 30, 2006 Due: Thursday, December 7, 2006Problem 11.1 Recall that a statistical function h is said to be continuous
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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Solutions - Problem Set 11 Fall 2006 Issued: Thursday, November 9, 2006 Due: Thursday, November 16, 2006Graded exercisesProblem 11.1 a) The functional h(F ) = F (a)
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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)
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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
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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.
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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
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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
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