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 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 (Patrick
Yuen, 1 hr/wk)
Course material Syllabus
(Ch.15) (A. van der Vaart); Solutions
to selected exercises
Further reading Book “Asymptotic
Statistics”, by A. W. van der Vaart, Cambridge University
press. (ISBN13: 9780521784504  ISBN10: 0521784506).
Course schedule Schedule,
Autumn 2023
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 alternative version of the final exam
(duration 3hrs, weight 100%) will be available.
