Advanced Probability and Statistical Inference I (BIOS 760)Fall 2008• COURSE DESCRIPTION (4 credit hours)The course introduces fundamental concepts of measure theory and probability measuretheory. Large sample theory in probability measure space is given, including a varietyof convergence results and the central limit theorems. The second part of the coursereviews a number of methods for point estimation, with particular attention to maximumlikelihood estimation and its related aspects.• MEETING TIME: 3:30–5:15 P.M., Tuesdays and Thursdays in room McG-G 1304• CLASS WEBSITE: http:\\www.bios.unc.edu\~kosorok\BIOS760.html• TEXTS FOR TEACHING– Lecture notes (downloadable from the website)– Theory of Point Estimation, Second Edition, Lehmann, E. and Casella, G., 1998– A Course in Large Sample Theory, Ferguson, T. S., 1996, reprinted 2002• INSTRUCTORDr. Michael R. KosorokOffice: 3101 McGavran-GreenbergEmail: [email protected]: 919-966-8107Office hours: 1–2 PM Tuesday, 2–3 PM Thursday, and some additional time slots byappointment• TEACHING ASSISTANTDr. Emil CorneaEmail: [email protected] hours: 11:00AM–noon on Monday, in room MHRC 3100; and noon–1:00PM onFriday, in room MHRC 0003.• GRADING SYSTEMFinal grade is based on the performance of bi-weekly homework, one midterm exam and1one final exam. The distribution is respectively 40%, 30% and 30%. The grades reportedwill be transformed to an “HPF” scale (H: 85–100; P: 70–84; LP: 60–69; F: 0–59).• MAIN TOPICS1. Distribution Theory (expected 1 week)– Basic concepts– Special distributions– Algebra and transformation of random variables– Multivariate normal distribution– Families of distributions2. Measure, Integration and Probability (expected 3 weeks)– Set theory and topology– Measure space– Construction of measure space– Measurable functions and integration– Product of measures–Fubini-Tonelli Theorem– Derivative of measures–Radon-Nikodym Theorem– Probability measure– Conditional probability and independence3. Large Sample Theory of Random Variables (expected 4 weeks)– Modes of convergence– Convergence in distribution– Limit theorems for summation of independent random variables– Limit theorems for summation of non-independent random variables–U-statisticsand Martingales– Some notation4. Point Estimation and Efficiency (expected 2 weeks)– Methods of point estimation– Cr´amer-Rao bound for parametric models– Information bound and efficient influence function2– Asymptotic efficiency bound: Le Cam’s lemmas5. Efficient Estimation: Maximum Likelihood Approach (expected 3 weeks)– Kullback-Leibler information– Consistency of maximum likelihood estimators– Asymptotic efficiency of maximum likelihood estimators– Computation of maximum likelihood estimators: EM algorithm– Nonparametric maximum likelihood estimation• OTHER INFORMATION– A number of problems are given at the end of each chapter of the lecture notes.Homework will be assigned from these problems and solutions will be posted inthe class webpage after grading. You are encouraged to work on the problems notassigned. Working in groups is not discouraged but plagiarism (copying) is strictlyprohibited. It is recommended that, if you wish to work in groups, that your firsttry the problems on your own before discussing them with the group. You may useideas you obtain from this group interaction for your solutions, but you must processthe solutions and write them in your own words. As stated before, copying problemsolutions is strictly prohibited.– Teaching tools will be mainly based on the use of the projector, sometimes with thehelp of the chalkboard or handouts. The slides for teaching can be downloaded fromthe webpage. You may wish to print out the slides with the blank note pages so thatadditional notes can be taken side by side.– Both midterm and final exams will be closed-book exams. The midterm will be heldduring a regular class slot.– Feel free to send me an email, give me a call or stop by during my office hours if youhave questions, comments, ideas or suggestions.– Work hard and never give