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Course: MATH 5061, Fall 2008
School: Washington University...
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University Washington Math5061 (Fall 2006) 1 Review 1. Basic probability theory (Chapter 1, LC) pdf, cdf, marginal, conditional, joint Bayes theorem mgf change-of-variable formula how to obtain moments from mgf properties: mgf for ax + b and x + y convergence in distribution, convergence in probability (1.8, LC) in probability in distribution hold for continuous function For an estimator, in...

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University Washington Math5061 (Fall 2006) 1 Review 1. Basic probability theory (Chapter 1, LC) pdf, cdf, marginal, conditional, joint Bayes theorem mgf change-of-variable formula how to obtain moments from mgf properties: mgf for ax + b and x + y convergence in distribution, convergence in probability (1.8, LC) in probability in distribution hold for continuous function For an estimator, in probability consistent MSE 0 consistent Slutskys theorem law of large numbers, central limit theorem exponential family (1.5 LC) canonical form, natural parameter, natural sucient statistics full-rank exponential family, curved exponential family mean and variance of the natural sucient statistics, E = A , V = A applications of Basus theorem location-scale family (1.4 LC) some property of normal distribution 2 2 2 2 N (1 , 1 ) + N (2 , 2 ) = N (1 + 2 , 1 + 2 + 21 2 ) N (0, 1)2 = 2 X Np (, ) AX + b (A + b, AAT ) 2. (Minimal) sucient, ancillary, and complete statistics (1.6 LC) Factorization theorem How to nd minimal sucient statistics? complete statistic minimal sucient statistics of full-rank exponential family complete sucient statistics of full-rank exponential family Basus theorem: complete sucient stat ancillary stat Washington University Math5061 (Fall 2006) 3. Unbiased estimation (Chapter 2) UMVU estimator (2.1, LC) 2 Rao-Blackwell theorem: improve unbiased estimators by conditioning on sucient statistic (2.1, LC) UMVUE must be a function of a complete sucient statistic if such a statistic exists (full-rank exponential family) (2.1, LC) How to nd a UMVUE when a complete sucient statistic exists? (2.1, LC) Directly solve the equation E ((T )) = g(). Conditioning an unbiased estimator on T application of Basus theorem for location-scale family (page 94, LC) Cramer-Rao lower bound inequality) (information (2.5) Fisher information I() = V ar(l ()) = E(l ()2 ) = E(l ()) What is the information for h()? What is the information in n i.i.d. observations? Fisher information for location-scale family Fisher information for = E(T ) in exponential family lower bound: var() ()2 /I(). If is unbiased, var() 1/I() Attainment of the bound for exponential family Another way to nd a UMVUE: an unbiased estimator that attains the bound 4. Equivariant estimation Pitmans estimator 5. Method of moments estimator easy to construct Often consistent 6. Least square estimators 7. Maximum likelihood estimator Consistency (6.3, LC) Delta method (6.3, LC) Computation Asymptotically normal and ecient (6.3, LC) Newton-Ralphson (page 453, LC) Washington University Math5061 (Fall 2006) Fisher scoring EM algorithm (page 457, LC) one-step estimator (page 454, LC) n-consistent 8. M-estimator (estimating equation) (page 483, LC) Fisher-consistent 3 How to derive the asymptotic normality? Taylor expansion Inuence function asymptotic normality of sample quantiles (e.g. ...

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Washington University in St. Louis - MATH - 5061
Washington University Math5061 (Fall 2006)1Homework 7: Due November 6, 20061. Suppose that Xn Binomial(n, p), then pn = Xn /n is the typical estimator of p. ^ Using the delta method, show that the variance stabilizing transformation is given by
Washington University in St. Louis - MATH - 5062
Washington University Math5062 (Spring 2007)1Homework 4, Due March 2, 20071. (Two sample problems) Suppose that we have two independent samples, X1 , . . . , Xn1 2 2 i.i.d. from N (1 , 1 ), and Y1 , . . . , Yn2 i.i.d. from N (2 , 2 ). Show that
Washington University in St. Louis - MATH - 5061
Washington University Math5061 (Fall 2006)1Homework 4: Due October 9, 20061. Suppose that X1 , . . . , Xn are a random sample from N (, 1). Find the Cramer-Rao lower bound for the variance of the unbiased estimators of 2 given by the information
Washington University in St. Louis - MATH - 5061
Washington University Math5061 (Fall 2006)1Homework 51 1. Let X1 , . . . , Xn be i.i.d. random variables having nite = E(X1 ) and = E(X1 ). Find a U-statistic that is an unbiased estimator of . 2. Consider the polynomial model Xi = 0 + 1 ti
Washington University in St. Louis - MATH - 5061
Washington University Math5061 (Fall 2006)1Homework 111. For Problem 3 in Homework 10, find the 90% HPD credible interval of . Note, if the exact HPD credible interval is difficult to calculate, you may find it using an normal approximation to t
Washington University in St. Louis - MATH - 5062
Washington University Math5061 (Spring 2007)1Homework 2, Due February 5, 20071. In class, we proved the Neyman-Pearson lemma for 0 &lt; &lt; 1. Verify the NeymanPearson lemma for = 0 and = 1. 2. Suppose X U nif orm(0, ), and we want to test H0 :
Washington University in St. Louis - MATH - 459
Math 459: Bayesian Statistics Solutions to HW 1Andrew Womack March 1, 2009Problem 1: The Monty Hall ProblemIntuitive SolutionThe information that we will condition on is the fact that Monty will only choose a lesser prize and will never choose t
Washington University in St. Louis - MATH - 459
Washington University Math459 (Spring 2009)1Homework 2: Due Friday February 6, 20091. Suppose y1 , . . . , yn form a random sample from N (, 1). Let the prior of be an improper at prior. Give the posterior distribution of |y1 , . . . , yn . 2.
Washington University in St. Louis - MATH - 5061
Washington University Math5061 (Fall 2006)1Homework 8: Due November 13, 20061. (Cauchy location family) Suppose that X1 , . . . , Xn are i.i.d. from the distribution for which the p.d.f. is 1 , - &lt; x &lt; . f (x) = [1 + (x - )2 ] (a) Find a n-cons
Washington University in St. Louis - MATH - 5061
Washington University Math5061 (Fall 2006)1Homework 6: Due October 30, 20061. Let X1 , X2 , . . . be i.i.d. with E(Xi ) = , V ar(Xi ) = 2 , and V ar(Xi2 ) = 2 &lt; . Let2 Sn1 = nni=1 (Xi Xn )2 .2 (a) Show that the distribution of Sn d
Washington University in St. Louis - MATH - 459
# Mixtures of Conjugate Priorslibrary(LearnBayes)# problem setup: estimate p in Bin(n,p)# a mixture of beta priorscurve(.5*dbeta(x, 6, 14) + .5*dbeta(x, 14, 6), from=0, to=1, xlab=&quot;P&quot;, ylab=&quot;Density&quot;)probs=c(.5,.5)beta.par1=c(6, 14)beta.p
Washington University in St. Louis - MATH - 459
## Section 6.7 Learning about a Normal Population from Grouped Data#library(LearnBayes) d=list(int.lo=c(-Inf,seq(66,74,by=2), int.hi=c(seq(66,74,by=2), Inf), f=c(14,30,49,70,33,15)y=c(rep(65,14),rep(67,30),rep(69,49),rep(71,70
Washington University in St. Louis - MATH - 5061
Washington University Math5061 (Fall 2006)1Homework 10: Due November 27, 20061. Suppose that we dene the loss function as L(, a) = | a|. Show that the Bayes estimator of is given by the posterior median. 2. Let X1 , . . . , Xk be independent r
Washington University in St. Louis - MATH - 5062
Washington University Math5061 (Spring 2007)1Homework 3, Due February 19, 20071. Let X1 , . . . , Xn be i.i.d. from N (, 1). For testing H0 : 1or 1 versus H1 : 1 &lt; &lt; 1, nd the level = 0.05 UMP test at n = 10. Specify the test function. 2. S
Washington University in St. Louis - MATH - 5062
Washington University Math5062 (Spring 2007)1Homework 6: Due Wednesday, April 181. Let X1 , . . . , Xn be i.i.d. binary random variables with = P r(X1 = 1). (a) Let the prior () be the c.d.f. of the beta distribution Beta(a, b). Find the Bayes
Washington University in St. Louis - MATH - 5062
Washington University Math5062 (Spring 2007)1Homework 5: Due Wednesday March 28, 20061. Problem 6.3.1 on page 428 in BD. 2. Problem 6.3.6 on page 430 in BD. 3. Let X1 , . . . , Xn be i.i.d. N (, 1) with 0. And we want to test H0 : = 0 versus
Washington University in St. Louis - MATH - 5061
Washington University Math5061 (Fall 2006)1Homework 9: Due November 20, 20061. Let X1 , . . . , Xn be i.i.d. from N (0, 2 ). One can nd an M-estimator of by minimizing the following objective function 1 nn(Xi /) + log i=1for some functio
Washington University in St. Louis - MATH - 5061
Washington University Math5061 (Fall 2006)1Homework 1: Due September 18, 20061. Problem 4 on page 87 of BD. Please also show whether Student's t-distribution belong to an exponential family or not. (BD - Bickel and Doksum (2001) 2. Let (X, Y ) h
Washington University in St. Louis - EPSC - 444
EPSc/EnSt 444 Review Paper Instructions Due in class December 4, 2008 LATE PAPERS WILL NOT BE ACCEPTED! The desired product of this assignment is a paper that reviews a specific topic in Environmental, Aqueous, or Sedimentary Geochemistry. You are ex
Washington University in St. Louis - CS - 260
CS/EE 260M Homework 1 Due Date: 1/27/20001. Mano 1-3. 2. Mano 1-4. 3. Mano 1-5. 4. Mano 1-8. 5. Mano 1-16. 6. Mano 1-24. 7. Mano 1-26. 8. Write a machine-language program similar to the one on page 1-9 of the course notes. Your program should add
Washington University in St. Louis - CS - 260
CS/EE 260M - Homework 3 Due Date: 2/10/20001. Mano 2-16. 2. Mano 2-20. 3. Mano 2-23. 4. Mano 2-24. 5. Mano 2-27. 6. Mano 2-29. Also, Use the schematic editor to enter the circuit for part (b) and simulate for all possible input values. Do both a fu
Washington University in St. Louis - CS - 260
CS/EE 260M - Homework 4 Due Date: 2/17/20001. Mano 3-1. 2. Mano 3-2. 3. Mano 3-4. 4. Mano 3-5. Hand in the schematic and simulation printouts. 5. Mano 3-7. 6. Mano 3-10. 7. Mano 3-12. 8. Mano 3-16. Use the schematic editor to enter the circuit and
Washington University in St. Louis - CS - 260
CS/EE 260M Homework 2 Due Date: 2/3/20001. Mano 2-4. 2. Mano 2-6. 3. Mano 2-8. 4. Mano 2-9. 5. Mano 2-11. 6. Mano 2-13. 7. Mano 2-14. 8. Mano 2-15. 9. Use the schematic editor to create a circuit with a single OR gate, plus input and output termin
Washington University in St. Louis - CS - 241
CS 241 Algorithms and Data StructuresJanuary 12, 1999Spring Semester, 1999Handout 1Course InformationFor lots of useful information about the class such as frequently asked questions, help with UNIX, o ce hours, the CS 241 suggestion box, and
Washington University in St. Louis - CS - 441
CS 441T 539T: Advanced AlgorithmsSyllabusJanuary 18, 2000Spring Semester, 2000Handout 2This syllabus will be adjusted as required to meet the needs of the class. See the course web page at http: classes.cec.wustl.edu cs441 for up-to-date infor
Washington University in St. Louis - CS - 441
CS 441T 539T: Advanced AlgorithmsJanuary 18, 2000Course InformationSpring Semester, 2000Handout 1Prerequisites: CS 241 and CS 201 or equivalent background. Time Place: Tuesday and Thursday 1:00-2:30pm, Crow 206 Professor: Dr. Sally A. Goldman
Washington University in St. Louis - CS - 201
CS201. Formal Foundations of Computer Science1Homework 2Due date: Feb. 10, 2000 As a reminder, you need not turn in the practice exercises. However, to provide strong encouragement to do them, one of them will be picked up as a probable closed-b
Washington University in St. Louis - CS - 241
CS 241 Algorithms and Data StructuresLab 1Fall Semester, 1997Due Date: September 9August 26, 1999B&quot; Portion:In order to earn a B&quot; on this lab you must successfully complete this portion of the lab. While a good part of the grade is attached
Washington University in St. Louis - CS - 577
CS/EE 577 Homework 3Due 11/9/001. (15 points) Consider a crossbar-based switch with 64 ports in which each IPP has a single queue and during each arbitration cycle, the IPPs contend for the output that the first cell in the queue is addressed to.
Washington University in St. Louis - CS - 577
CS/EE 577 Homework 1Due 9/21/001. How many pixels per second must be sent for a video signal with a resolution of 640 by 480 pixels per frame and a frame rate of 30 frames per second? What data rate is needed to support this at 24 bits per pixel?
Washington University in St. Louis - CS - 260
CS/EE 260 - Homework 7Due 3/23/20001. Enter schematics for the SR master-slave flip flop and for the D edge-triggered flip flop in the schematic editor, using just NAND gates and inverters. Simulate the flip flops with the input waveforms shown be
Washington University in St. Louis - CS - 260
CS/EE 260 - Homework 6Due 3/16/20001. Use VHDL to design the 4 bit arithmetic unit specified in problem 4 of homework 5 (you may borrow from the posted solution, if you wish). Use a dataflow description to define basic building blocks for each com
Washington University in St. Louis - CS - 260
CS/EE 260 - Homework 9 SolutionsDue 4/6/20001. Consider the synchronous ripple carry counter on page 5-8 of the notes. Assume that the D flip flops have a setup time requirement of 2 ns and that the gates in the circuit all have a propagation dela
Washington University in St. Louis - CS - 260
CS/EE 260 - Homework 7Due 4/2/20011. (20 points) A 4 bit twisted ring counter is a sequential circuit which produces the following sequence of output values: 0000, 1000, 1100, 1110, 1111, 0111, 0011, 0001 and then repeats. Design a circuit for a 4
Washington University in St. Louis - CS - 533
PROGRAM A FAQ (FREQUENTLY ASKED QUESTIONS)-Last Updated: Sun, 1015 AM, Feb. 20, 2000INDEX-Q1) When I tried to read man pages in /home/cec/class/cs533/projects/proj1/man1, I got error messages about not being able to find the fi
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PROGRAM A FAQ (FREQUENTLY ASKED QUESTIONS) -Last Updated: Fri, 0845 AM, Feb. 9, 2001INDEX-Q1) I just lost all of my files. What can I do to recover them?Q2) How can client and / or server tell each other whi
Washington University in St. Louis - CS - 533
CS 533S - HOMEWORK 5 SOLUTION Page 1 -Version 4, Apr. 26, 2001Problem 2-a) TCP has gone through many changes in the last 15-20 years. Furthermore, there is no &quot;standard&quot; T
Washington University in St. Louis - K - 30
SCHOLAR POWERThe Clinical Research Training Center Newsletter Volume II, Issue 1 Spring 2008In This Issue Scholar Spotlight CTSA News Upcoming Events Program News &amp; NotesScholar Spotlight - Stephanie Lutter, MDDr. Stephanie Lutters interest in
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COMMENTARYA Social Science Perspective on Gifts to Physicians From IndustryJason Dana, MS George Loewenstein, PhD cal or professional interest clashes with financial self-interest, a situation that arises commonly in medical practice. When physici
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REVIEW ARTICLEPharm Med 2008; 22 (2): 75-84 1178-2595/08/0002-0075/\$48.00/0 2008 Adis Data Information BV. All rights reserved.Biobanking Human Biological MaterialsIssues Surrounding the Collection of Samples for Use in Future ResearchMarshall
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REVIEW ARTICLEPharm Med 2008; 22 (2): 75-84 1178-2595/08/0002-0075/\$48.00/0 2008 Adis Data Information BV. All rights reserved.Biobanking Human Biological MaterialsIssues Surrounding the Collection of Samples for Use in Future ResearchMarshall
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False Hopes and Best Data: Consent to Research and the Therapeutic Misconception Paul S. Appelbaum; Loren H. Roth; Charles W. Lidz; Paul Benson; William Winslade The Hastings Center Report, Vol. 17, No. 2. (Apr., 1987), pp. 20-24.Stable URL: http:/l
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S OUND ING B OA RDSounding Board N ONFINANCIAL C ONFLICTS OF I NTEREST IN R ESEARCHIN 1963, before the advent of institutional review boards (IRBs), I was a young academic physician studying the regulation of sodium excretion by the kidneys. I
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www.emhr.net A FRAMEWORK FOR ANALYZING ETHICS CASES*James M. DuBoisProfessional ethics is an applied field. It is not primarily about theory and rote knowledge, but about shaping our character and fostering good ethical decisions. In what follows,
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16Case ReportWELKER, MCCUE, Authorship versus Credit for Participation in ResearchAuthorship versus Credit for Participation in Research: A Case Study of Potential Ethical Dilemmas Created by Technical Tools Used by Researchers and Claims for Au
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16Case ReportWELKER, MCCUE, Authorship versus Credit for Participation in ResearchAuthorship versus Credit for Participation in Research: A Case Study of Potential Ethical Dilemmas Created by Technical Tools Used by Researchers and Claims for Au
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HEALTH POLICY AND ETHICS FORUM French collaborative study group. Am J Epidemiol. 1995;142:13301337. 25. Consensus statement. Science, ethics, and the future of research into maternal infant transmission of HIV-1. Lancet. 1999;353:832835. 26. Miroc
Washington University in St. Louis - K - 30
Washington University in St. Louis - K - 30
Epidemiologic Reviews Copyright 2000 by The Johns Hopkins University School of Hygiene and Public Health All rights reservedVol. 22, No. 1 Printed in U.S.A.Ethics in Epidemiology at the End of the 20th Century: Ethics, Values, and Mission Statem
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PERSPE C T I V Esubjects or objects? prisoners and human experimentationSubjects or Objects? Prisoners and Human ExperimentationBarron H. Lerner, M.D., Ph.D.During the 1950s, inmates at what was then called Holmesburg Prison, in Philadelphia
Washington University in St. Louis - K - 30
Washington University in St. Louis - K - 30
INFORMED CONSENTmust be balanced against the interests of other patients and of society in maintaining a productive and efficient healthcare system. Accordingly, institutional policies must consider what is fair and reasonable to require of health
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SPECIAL COMMUNICATIONEthics of Phase 1 Oncology StudiesReexamining the Arguments and DataManish Agrawal, MD Ezekiel J. Emanuel, MD, PhD ments in treatments for individual cancers, 50% of people diagnosed as having cancer, more than 550 000 annual
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EDICINE AND PuBLIC ISSUESPlacebo-Controlled Trials and Active-Control Trials in the Evaluation of New TreatmentsPart 1: Ethical and Scientific IssuesRobert Temple, MD, and Susan S. Ellenberg, PhD In recent years, several authors have argued that
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Informed consent in pediatric clinical trialsRaymond C. Barfielda and Christopher ChurchbPurpose of review Clinical research in children is increasing. Concerns have been raised about both the inclusion and the exclusion of children in such resear
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REVIEWScope and Impact of Financial Conflicts of Interest in Biomedical ResearchA Systematic ReviewJustin E. Bekelman, AB Yan Li, MPhil Cary P. Gross, MD research in the United States increased dramatically in the last 2 decades. Industry's share
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