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### stat210a_fall07_hw7

Course: STAT 210a, Fall 2007
School: Berkeley
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Word Count: 615

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Berkeley UC 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 ). (a) Show that (X ) = C n n i=1 |Xi | Due: Thursday, October 18 is a consistent estimator of if and only if C = 1 n n 2 i=1 Xi . /2. (b) Show that...

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Berkeley UC Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 7 Fall 2007 Issued: Thursday, October 25 Reading: Keener: Chapter 10, 11. B & D: Chapter 5. Problem 7.1 Suppose that X1 , . . . , Xn are i.i.d. N (0, 2 ). (a) Show that (X ) = C n n i=1 |Xi | Due: Thursday, October 18 is a consistent estimator of if and only if C = 1 n n 2 i=1 Xi . /2. (b) Show that the MLE of is given by M LE (X ) = totic relative eciency of with c = Determine the asymp- /2 compared to the MLE M LE . Problem 7.2 Suppose that X1 , . . . Xn are i.i.d from the normal location model N (, 1), and that we wish to estimate the critical or cuto value ga () = P[X1 a], where a R is some xed number. (a) Let denote the CDF of the standard normal distribution. Show that the estimator n (X ) = n (a Xn ) n1 d n(n ga ()) N (0, 2 (a )), where is an unbiased estimator of ga (). Prove that is the PDF of the standard normal. (b) If we had doubts about the assumption of normality, we might prefer to use the non1 parametric estimator n (X ) = n # {Xi a}, that simply counts the fraction of Xi that are less than or equal to a. Prove that d n(n ga ()) N (0, (a )[1 (a )]) . (c) Compute the ARE of and . If Xi truly are Gaussian, how much is lost by the non-parametric estimator ? Problem 7.3 Suppose that Xi , i = 1, . . . , n are i.i.d. samples from a normal location model N (, 1), and that we are interested in estimating the quantity 1/. In order to do so, we use the estimator 1 (X ) = 1/Xn where Xn = n n Xi is the sample mean. i=1 (a) Show that is asymptotically normalviz.: 1 d n 1/Xn 1/ N (0, 1/4 ). (b) Show that the expectation E[1/Xn ] fails to exist for all n. Why does this not contradict the result of part (a)? Problem (Censored 7.4 Poisson) Suppose that Yi , i = 1, . . . , n are drawn i.i.d. from a Poi() distribution, but that we only observe the censored quantities Xi = 0 if Yi = 0 1 if Yi > 0. (a) Under what conditions does the ML estimate M LE of based on (X1 , . . . , Xn ) exist? Compute the probability that the MLE exists as a function of and n. What happens as n +? (b) Dene the modied estimator (X ) = Prove that M LE (X ) 1 when the MLE exists otherwise. d n( ) N (0, exp() 1). Problem 7.5 Suppose that we have n i.i.d. samples {(Xi , Yi )}n from the bivariate normal distribution i=1 2 2 with zero means, variances X and Y and correlation coecient (1, 1). 2 2 (a) First supposing that X and Y are known constants, show that the MLE M LE satises d n (M LE ) N 0, (1 2 )2 1 + 2 . 2 2 Now suppose that X and Y , in addition to , are unknown parameters. In the following 2 2 sub-problems, we analyze the asymptotic behavior of the MLE (X , Y , ). (Note that 2 2 will not be the same, in general, as the MLE M LE from part (a) in which X and Y are assumed to be known.) 2 2 (b) Compute the 3 3 Fisher information matrix associated with = (X , Y , ). (c) Verify that the inverse of the Fisher information matrix computed in (b) is given by 4 22 2 2X 22 X Y (1 2 )X 22 4 2 2Y (1 2 )Y [I ()]1 = 22 X Y 2 2 (1 2 )X (1 2 )Y (1 2 )2 (Hint: You do not need to try and compute this matrix inverse explicitly; simply compute the Fisher information matrix, and verify by matrix multiplication this form 2 2 of its inverse.) Use this result to derive asymptotic distributions for n(X X ), 2 2 n(Y Y ), and n( ). (d) Compute the asymptotic relative eciency (ARE) of M LE from (a) compared to . 2 2 What does this tell you about the value of knowing X and Y ? 2
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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
Berkeley - STAT - 210a
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 11 Fall 2007 Issued: Monday, November 19 Reading: Keener: Chapter 22; B &amp; D: Chapter 5 Problem 11.1 2 2 Consider the simple binary hypothesis test H0 : X N
Berkeley - STAT - 210a
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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 exercisesProblem 2.1 From the distribution of X, we have:nP (X
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, 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 4 Fall 2006 Issued: Thursday, September 21, 2006 Due: Thursday, September 28, 2006Graded exercisesProblem 4.1 First, notice that we can rewrite: p(x; ) =
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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 5 Fall 2006 Issued: Thursday, September 28, 2006 Due: Thursday, October 5, 2006A useful definition for this problem set: Definition: An equalizer procedur
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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-) )
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
Berkeley - STAT - 210a
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
Berkeley - STAT - 210a
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
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
Berkeley - STAT - 210a
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)
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
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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
Berkeley - STAT - 210a
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