# Asymptotics

DBP 12 Jan 2016

 Instructor: David Pollard When: Tues 10:30-12:00 + working sessions TBA Where: 24 Hillhouse Office hours: part of working sessions TA: David Pollard Other: courses taught by DP in previous years Short description: A careful study of some standard asymptotic techniques in statistics and econometrics, and their modern refinements. Topics selected from classical likelihood theory and M-estimation; empirical process methods; concentration inequalities; semiparametric models; local asymptotic normality; concepts of efficiency. Students will be expected to read research papers and make presentations in class. Prerequisites: knowledge of probability at the level of STAT 600b. Intended audience: Students who have had some exposure to measure theoretic probability and who need to understand some asymptotic theory at the research level. Resources: Go to Resources on the Stat 618 web site at Classes V2 Source material: Once upon a time I decided to update the empirical process material in my 1984 and 1990 books. I needed some brief document that I could, in good conscience, recommend to students who wanted to understand some of the main ideas in the subject, including improvements made in the last twenty years. Unfortunately, the longer I labored the further I seemed from my objective. A few years ago I decided I might actually finish if I split the Asymptopia manuscript into two parts. Currently I am working on the empirical process bit, which has acquired the temporary working title MiniEmpirical. Some chapters have reached a reasonably complete form. Those chapters are in the Mini subdirectory. I would also like to show how these Mini tools can be used in statistical asymptotic theory. Some of the examples will come from the dreaded Asymptopia manuscript. Grading: The final grade will be determined by a student's participation in the working sessions, where small groups will be asked to supply details for some of the sketched arguments or to explain selected applications from the literature. I have used this method a lot in recent years, for courses such as Stat 603 in 2010 and for various incarnations of Stat 690. Here is the plan. Once a week I will try to give the main ideas for some important topic. Initially I will provide a lot of detail. As the semester progresses I will leave more of the argument for you to explain to me in the weekly working sessions. Meanwhile I intend to work assiduously, trying to turn some of the messy drafts into helpful documents for your consideration. Of course you are welcome to talk with me to clarify your thinking before the working sessions. Topics: Tentative list. The actual material covered will depend, in part, on the backgrounds of students in the class. Multivariate normal distribution; tail bounds; concentration of Lipschitz functionals; Borell's inequality; Slepian's inequality. [Mini chapter Gaussian] Exponential tail bounds via moment generating functions: Hoeffding, Bennett with some mention of martingale analogs [Mini chapters Basic and BasicMG] Chaining, which is just a fancy application of the triangle inequality. [Mini chapter Chaining] Symmetrization, and why symmetry need not be the reason for symmetrizing. [Soon to be Mini chapter Symmetrization] VC classes and combinatorial dimension. [Soon to be Mini chapters on VC sets and fat shattering] M-esimators. Stochastic equicontinuity. [Asymptopia] (maybe) Something about differentiabilty in quadratic mean and local asymptotic normality. [Asymptopia]