Thomas Stieltjes Institute for Mathematics

Stochastics 1997/98

Title: Course Large Deviations and
Course Introductory Course to Time Series Analysis
Organisation AIO-netwerk Stochastiek zie ook:
(A.W. van der Vaart (), tel: 020-4447697, J.H.J. Einmahl ( ), tel: 040-2472499).
Lecturers: Frank den Hollander (first course)
Thomas Mikosch (second course)
Time and place: Fall 97
We have planned the dates:
September 5 and 19,
October 3, 17 and 31*,
November 14 and 28,
December 12.

(*The probability session of 31 October will be moved to 21 November, in Lunteren.)

The morning session will be probability and the afternoon session will be statistics. As usual the courses consist of eight lectures of two hours each, given on eight Fridays. The courses will probably be given in Utrecht, in the Academie-gebouw, Domplein, room 10. Morning and afternoon sessions are planned for 10.45-12.30 and 13.30-15.15, respectively. A final announcement will follow.
Abstract: Frank den Hollander (KUN) ()

The theory of large deviations deals with the computation of `small' probabilities, away from the central limit theorem. The reason for studying these probabilities is that they come up in a wide variety of situations, e.g. the study of functionals of random processes, the calculation of tail properties for sums of weakly dependent random variables, error estimates, etc. Large deviation theory has proved to be the crucial tool in handling many questions in statistics, probability and statistical mechanics.

The course will have three parts:

  1. Exposition of the foundations of the theory, illustrated for i.i.d. sequences and Markov sequences (3 lectures).
  2. Description of the general framework of the theory: large deviation principle, Varadhan's theorem, contraction principle, etc. (2 lectures).
  3. Discussion of some applications: hypothesis testing, conditioned limit theorems, random walks in random media, Curie-Weiss model for ferromagnetism, polymers.

J.D. Deuschel and D.W. Stroock, Large Deviations, Academic Press, Boston, 1989.
J.A. Bucklew, Large Deviation Techniques in Decision, Simulation, and Estimation, Wiley, New York, 1990.
A. Dembo and O. Zeitouni, Large Deviation Techniques and Applications, Jones and Bartlett Publishers, Boston, 1993.

Abstract: Thomas Mikosch (RUG) ()

By now, time series analysis is one of the well established theories in probability theory and mathematical statistics with applications in various fields such as hydrology, meteorology, finance, astronomy, economics. The stochastic model underlying it is mostly assumed to be a stationary discrete--time process. The aim of the analysis is to detect the dependence structure in the series, to fit an appriate model and to predict future values in the series. This is done in the time or frequency domain. The first method is based on considerations of the autocorrelations and autocovariances. The second one uses Fourier analytic methods; it is also called spectral analysis.

The analysis of time series is mostly based on Hilbert space techniques. They allow one to represent a stationary process as a stochastic integral with trigonometric functions as integrands. Thus a stationary process, to a first approximation, can be understood as a sinosoid with random coefficients. The order of magnitude of those coefficients is discribed by the spectral distribution function and its density. The course will give a gentle introduction to modelling of time series both in the frequency and time domain, stressing more the understanding of the models than the discussion of technical details. Classical ARMA processes (=autoregressive moving average processes) allow us to give insight into the structure of more complicated models. ARMA models will serve as prime examples for developing a spectral and time domain analysis which is accompanied by an introduction to the statistical tools for fitting data. Prediction of times series is one of the main issues and will be described in detail.

In addition to the basics of classical time series modelling the following recent issues of interest will be discussed:

Recommended references are:
Brockwell, P.J. and Davis, R.A. (1991) {Time Series: Theory and Methods,2nd~edition. Springer, New York.
Brockwell, P.J. and Davis, R.A. (1996) { Introduction to Time Series and Forecasting.} Springer, New York.
The first reference is an introduction to time series analysis for a mathematically oriented person, the second one is for the more applied person who is interested in how things work, but does not ask why.

Title: Course Statistical Modelling of Medical Data and
Brownian Motion and the Funtional CLT
Organisation AIO-netwerk Stochastiek (A.W. van der Vaart (), tel: 020-4447697, J.H.J. Einmahl ( ), tel: 040-2472499).
Lecturers: Hans van Houwelingen (first course)
Gerard Hooghiemstra (second course)
Time and place: Spring 97
We have planned the dates:
January 23,
February 6 and 20,
March 6 and 20,
April 1 and 25
May 2.

The morning session will be statistics and the afternoon session will be probability. As usual the courses consist of eight lectures of two hours each, given on eight Fridays. The courses will probably be given in Utrecht, in the Academie-gebouw, Domplein, room 10. Morning and afternoon sessions are planned for 10.45-12.30 and 13.30-15.15, respectively. A final announcement will follow.

Possible late changes in the schedule will be communicated by e-mail and are accessible via the WWW page of A.W. van der Vaart (
Contact: L. de Haan (EUR), 010 - 408 1258, ()
A.W. van der Vaart (VUA), 020 - 444 7697, ()
Abstract: Hans van Houwelingen (RUL) ()

Medicine is one of the major fields of applications of statistics. In this course a review will be given of statistical models that are frequently used in medical research, such as logistic regression, survival analysis and models for pedigree data. At the departement of Medical Statistics at Leiden University active research is taking place on the further development of these kind of models. Topics of redearch are the goodness-of-fit of models and their predictive value in new data sets obtained in different populations. In the course some results of this research wil be presented. The selected references below give some flavour of the topics to be discussed.

J.C. van Houwelingen, S. le Cessie, Predictive value of statistical models, Statistics in Medicine 9, 1303-1325, 1990.
S. le Cessie, H.C. van Houwelingen, Testing the fit of a regression model via score tests in random effects models, Biometrics 51, 600-614, 1995
H.C. van Houwelingen, J. Thorogood, Construction, validation and updating of a prognostic model for kidney graft survival, Statistics in Medicine 14, 1999-2008, 1995
P.J.M. Verweij, H.C. van Houwelingen, Cross-validation in survvial analysis, Statistics in Medicine 12, 2305-2314, 1993
J.J. Howing-Duistermaat, B.H.F. Derkx, F.R. Rosendaal, H.C. van Houwelingen, Testing familial aggregation, Biometrics 51, 1292-1301, 1995

Abstract: Gerard Hooghiemstra (TUD) ()

This course is on Brownian motion and the Functional Central Limit Theorem. (FCLT).
We start with an introduction to Brownian motion involving: a full proof of continuity of paths, reflection principle, calculation of distributions of funtionals (by various methods, including exponential martingales). As reference we take the book of Freedman.
In connection with the above topic I intend to treat two FCLT's with applications:

Freedman, D. Brownian Motion and Diffusion. Holden Day, 1971, ISBN 0-8162-3024-2
Pollard, D. Convergence of Stochastic Processes. Springer Verlag , 1984, ISBN 0-387-90990-7
Groeneboom, P., Hooghiemstra, G. and Lopuhaa, H. P. Asymptotic normality of the L_1 error of the Grenander estimator. Report 97-01, TWI, TUDELFT.

Title: Course Measure theory, martingales and stochastics integrals.
Organiser: L. de Haan ()
Time and place: October-December 1997, EUR
Aim: The purpose is to enable PhD students in e.g. finance to deal with martingales and stochastic integrals
Contents: As above. There are extensive exercise sessions
Literature: David Williams: Probability with martingales. Cambridge Univ. Press 1991.
Prerequisites: regular course in probability and statistics at the intermediate level (as in the program of Econometrics).

Title: Course Semiparametric Models
Lecturers: prof. dr. C.A.J. Klaassen () and
dr. A.J. van Es ()
Time and place: Department of Mathematics, University of Amsterdam
Plantage Muidergracht 24, Amsterdam
The exact time is not yet planned, but it will be in our second trimester (december - march)
Aim: To get insight in the principles of semiparametric statistics
Contents: In classical statistics models with a (finite dimensional) parameter are studied. These models can be extended to so called semiparametric models by adding infinite dimensional parameters. The simplest example is the extension of the location model, where the distribution is assumed to be normal, to a model where the distribution is still assumed to be symmetric, but otherwise arbitrary. The theory that will be reviewed has been developed during the last fifteen years. This theory will be illustrated by the above mentioned symmetric location model, as well as by the Cox proportional hazards model (applied in Medical Statistics) and the regression model of Engle, Granger, Rice and Weis (Econometrics).
Literature: (optional) Bickel, P., Klaassen, C.A.J., Ritov. Y. and J.A. Wellner (1993), Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins, Baltimore and London
Prerequisites: Measure theoretic probability theory, basic statistics and some functional analysis.

Title: Course Analysis of Variance and Design of Experiments
Lecturers: E.E.M. van Berkum and ir. R.W.M. Jeurissen. address of Van Berkum:
University of Technology HG 9.01
P.O. Box 513
5600 MB Eindhoven
tel:040-2472903 ( )
Time: spring 1998
Aim: A student who attended the course must be able to
  • formulate a factorial model with the appropriate assumptions for a given problem; the model can be very general (e.g. with random and/or fixed factors and nested factors)
  • apply analysis of variance in factorial models and derive the probability distributions of the appropriate sums of squares
  • give designs for all kinds of practical problems
  • apply the general theory(D-optimality, G-optimality) of design of experiments in regression models.
Structure: The course is given in 9 weeks. In each week 4 lectures of 45 minutes each are given(in dutch). Moreover, there is a tutorial of 2 hours each week. There is an exam of 3 hours.
Contents: The main topics of the course are
  • factorial models (in general; fixed and/or random effects; nested and/or crossed factors)
  • analysis of variance including the derivation of the probability distributions of the sums of squares involved
  • general factorial designs
  • (fractional) designs (in blocks) for models where all factors have two levels or all factors have three levels
  • designs for fitting response surfaces (e.g. central composite designs)
  • block designs (e.g. latin squares)
  • Taguchi's distributions to design of experiments
  • D-optimal and G-optimal designs for regression models
  • Montgomery, D.C. Design and analysis of experiments, fourth edition. Wiley, New York.
  • Syllabus 2484. Variantieanalyse en proefopzetten. (english version on request available; contact E. van Berkum)
Prerequisites: It is necessary to know the basic principles of statistics (estimation, confidence intervals, hypothesis testing) and to have some knowledge of regression analysis.
Examination: It is possible to solve a practical problem in stead of 25% of the exam.
Title: Course Mathematical Statistics 2
Lecturers: dr. J.H.J. Einmahl ()
and dr. A. Di Bucchianico (tel:040-2472902 ()
University of Technology Eindhoven HG 9.02
P.O. Box 513
5600 MB Eindhoven.
Time: autumn 1997.
Aim: Estimation is a very important issue in statistics. In basic statistical courses much attention is paid to estimation of parameters (including confidence intervals). In practice (e.g. statistical quality control) this is very often not enough. One needs to estimate functions of parameters (e.g. quantiles or the probability that a random variable exceeds a certain specified value). The aim of this course is to give an introduction with respect to this kind of topic. Key words are "tolerance intervals" and "density estimation". Moreover this course gives an introduction on "Empirical processes", a modern tool in statistics. The teachers want to give a solid theoretical background for the use of statistics in practice.
Structure: The course is given in 9 weeks. In each week 3 lectures of 45 minutes each are given. In these lectures tutorials are included.
Contents: In the first two weeks estimation of functions of parameters and tolerance intervals are discussed. In the next 5 weeks an introduction is given on empirical processes. This theory will be applied on the asymptotic distribution of some non-parametric tests and on tolerance intervals. Q-Q plots will also be discussed. As an illustration of the use of Q-Q plots among other things tests on normality (e.g. Shapiro-Wilk) will be treated. In the last two weeks density estimation will be discussed with emphasis on kernel estimators.
Literature: A syllabus (in english) will be used.
Prerequisites: It is necessary to know the basic principles of statistics (estimation, confidence intervals, hypothesis testing).
Examination: There will be an oral exam.

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