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 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
socalled 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 Chisquare tests, Moment
estimators, Mestimators (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 theoremproof 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 sigmaalgebras,
measurable functions, measures, sigmaadditivity,
integration, monotone limits, etc, should not be wholly
unknown. For those participants who feel underequipped
measuretheoretically, 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 (Shreehari
Bodas and Leihao
Chen, 1 hr/wk)
Course material Syllabus
(Ch.15) (A. van der Vaart); Solutions
to selected exercises; Example Midterm exam (Oct 23)
Further reading Book “Asymptotic
Statistics”, by A. W. van der Vaart, Cambridge University
press. (ISBN13: 9780521784504  ISBN10: 0521784506).
Course schedule Schedule,
Fall 2024
Examination Written midterm exam
(duration 2 hrs, weight 50%); written final exam (duration
2 hrs, weight 50%); retake 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.
