1) Prerequisites
It is assumed that participants in the course have 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 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 (concurrent) course in Measure Theoretic Probability is highly recommended.
2) Aim of the course
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 desired uncertainty level, and the more data, the better the approximation. 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.
3) Rules about Homework/Exam
Written midterm exam (duration 2 hrs, weight 50%); written final exam (duration 2 hrs, weight 50%); re-sit exam (duration 3 hrs, 100%). For those who do not have a (satisfactory) grade for the midterm exam, an extended version of the final exam (duration 3hrs, weight 100%) will be available. A grade of 5 or higher on the final exam (both for 2hr and 3hr version) is a requirement to pass, otherwise the re-sit exam has to be taken.
4) Lecture notes/Literature
Lecture notes will be provided. Background literature: the book, "Asymptotic Statistics", by A. W. van der Vaart, Cambridge University press.
5) Lecturers
Bas Kleijn (lectures) and Georg Meyl (exercise classes)
- Docent: Bas Kleijn