Korteweg-de Vries Institute for Mathematics

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 Asymptotic Statistics, autumn 2023 Curriculum MasterMath Study burden 8 ECTS Goal Learn to study statistical procedures from an asymptotic point of view. Motivation In Asymptotic Statistics we study the asymptotic behaviour of (aspects of) statistical procedures. Here “asymptotic” means that we study limiting behaviour as the number of observations tends to infinity. A first important reason for doing this is that in many cases it is very hard, if not impossible to derive for instance exact distributions of test statistics for fixed sample sizes. Asymptotic results are often easier to obtain. These can then be used to construct tests, or confidence regions that approximately have the correct uncertainty level. Similarly, determining estimators or other procedures that are optimal in a specific sense, for instance in the sense of minimal mean squared error or variance, is often not possible if the number of observations is fixed. Using asymptotic results is it however in many cases possible to exhibit procedures that are asymptotically optimal. In this course we begin by treating the mathematical machinery from probability theory that is necessary to formulate and prove the statements of asymptotic statistics. Important are the various notions of stochastic convergence and their relations, the law of large numbers and the central limit theorem, the multivariate normal distribution, and the so-called delta method. We will use these tools to study the asymptotic behaviour of statistical procedures. Content The course starts with a review of various concepts of stochastic convergence (e.g. convergence in probability or in distribution) and properties of the multivariate normal distribution. Then the asymptotic properties of various statistical procedures are studied, including Chi-square tests, Moment estimators, M-estimators (including MLE). The examples are chosen according to importance in practical applications, and the theory is motivated by practical relevance, but the subjects are presented in theorem-proof form. Prerequisites It is assumed that participants in the course have, at the least, some knowledge of the basic concepts in statistics: estimation, testing and confidence sets; the definitions of moment estimators and the maximum likelihood estimator; the law of large numbers and the central limit theorem; normal, exponential, gamma, binomial, poisson families of distributions etc. Furthermore, at least a passing familiarity with measure theory is indispensable at the beginning of the course: concepts like sigma-algebras, measurable functions, measures, sigma-additivity, integration, monotone limits, etc, should not be wholly unknown. For those participants who feel under-equipped measure-theoretically, the (simultaneous) course in Measure Theoretic Probability is highly recommended. Registration Registration with MasterMath is required Teaching Lectures (Bas Kleijn, 2 hrs/wk) and Exercise classes (Patrick Yuen, 1 hr/wk) Course material Syllabus (Ch.1-5) (A. van der Vaart); Solutions to selected exercises Further reading Book “Asymptotic Statistics”, by A. W. van der Vaart, Cambridge University press. (ISBN-13: 9780521784504 | ISBN-10: 0521784506). Course schedule Schedule, Autumn 2023 Examination Written midterm exam (duration 2 hrs, weight 50%); written final exam (duration 2 hrs, weight 50%); re-take exam (duration 3 hrs, 100%). For those who do not have a (satisfactory) grade for the midterm exam, an alternative version of the final exam (duration 3hrs, weight 100%) will be available.