MaPhySto
Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

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Funded by The Danish National Research Foundation

MaPhySto Summer school on

Empirical Processes

From Monday, August 9, 1999 to Friday, August 20, 1999 MaPhySto organized a summer school on Empirical Processes. There were 4 series of main lectures with the following content:

1. Uniform Central Limit Theorems by R. M. Dudley (MIT)

   Summary: One of the main topics of empirical process theory is asymptotic normality of suitably normalized partial sums uniformly over classes of sets and functions. For the uniform convergence to hold there must be a limiting Gaussian process with sample continuity and boundedness. First, these properties of Gaussian processes will be treated in terms of metric entropy and the Talagrand-Fernique majorizing measure theorem. Then, combinatorial properties sufficient for uniform central limit theorems uniformly over all underlying probability measures will be studied. A good property for families of sets is finiteness of the Vapnik-Chervonenkis or VC index, also studied in computer learning theory. The VC property has various extensions to families of functions. Another useful property is bracketing, where families of functions are covered by unions of brackets [f_i,g_i], where [f,g] is the set of measurable functions h with f =< h =< g, and there are suitable bounds of the number of brackets in relation to some distance between f_i and g_i in mean or mean square.
   Some of the lectures was based on parts of a book by the author, also called Uniform Central Limit Theorems published by Cambridge University Press.

   The notes for these lectures may be fetched in various formats.

2. Empirical Processes at Work in Statistics by A.W. Van der Vaart (Amsterdam) & J.A. Wellner (Seattle)

   Summary: The lectures of Van der Vaart and Wellner will focus on the use of empirical process methods in dealing with a variety of questions and problems in statistics. Our examples and applications will be drawn from problems concerning semi-parametric models and non-parametric estimation for inverse problems. We will begin with a review of bounds for suprema of empirical processes, and will then discuss uses of these bounds in establishing:

   1. consistency of M- and Z-estimators;
   2. rates of convergence;
   3. convergence in distribution of maximum likelihood, sieved and penalized maximum likelihood estimators

3. Empirical and Partial-sum Processes Revisited as Random Measure Processes by P. Gänssler (Munich)

   Summary: In a general framework of so-called random measure processes (RMP's) we present uniform laws of large numbers (ULLN) and functional central limit theorems (FCLT) for RMP's yielding known and also new results for empirical processes and for so-called smoothed empirical processes based on data in general sample spaces. At the same time one obtains results for partial-sum processes with either fixed or random locations. Proofs are based on tools from modern empirical process theory as presented e.g. in Van der Vaart and Wellner [(1996): Weak Convergence and Empirical Processes; Springer Series in Statistics]. Our presentation will be also guided by showing up some aspects of the development of empirical process theory from its classical origin up to the present which offers now a wide variety of applications in statistics as demonstrated e.g. in Part 3 of Van der Vaart and Wellner [1996].

4. Convergence in Law of Random Elements and Sets by J. Hoffmann-Jørgensen (Aarhus)

   Summary: The classical definition of convergence in law of random elements is founded on convergence of the upper expectation of continuous functions. This concept has served very well in the theory of law convergence of empirical processes when the underlying topological space is metrizable or at least has sufficiently many continuous functions. However, in the context of law convergence of random sets associated to empirical processes (e.g. zero-sets or max-sets), the concept trivializes because the natural topology (the upper Fell topology) has no non-constant continuous functions. In the lectures I shall present a new concept of law convergence (convergence in Borel law) which coincides with the classical definition in ``nice'' topological spaces, and I shall demonstrate how this concept provides sensible limit theorems for random sets. In particular, we shall derive new and old results for law convergence of a certain class of estimators (J-estimators) which includes zero estimators and maximum estimators.

Final Schedule

Lectures

The Summer School took place in Aarhus at the Department of Mathematical Sciences, University of Aarhus..

Legend:

AB:= A. Bufetov
AVV:= A. Van der Vaart
DM:= D. Marinucci
FB:= F. Bravo
GP:= G. Peskir
JAW:= J.A. Wellner
JHJ:= J. Hoffmann-Jørgensen
LM:= L. Menneteau
MBH:= M.B. Hansen
RH:= R. Huntsinger
MP:= M. Piccioni
MS:=M. Scavino
OEBN:= O.E. Barndorff-Nielsen
PG:= P. Gänssler
RB:= R. Bilba
RMC:= R. McCrorie
RMD:= R.M. Dudley
SEG:= S.E. Graversen
SVG:= S. van der Geer
TS: T. Schreiber
VD:= V. Dobric
VDP:= V. de la Pena
WS:= W. Stute

Pictures from the Summer School!

More Information

Jointly with Centre for Analytical Finance MaPhySto is organizing an Instructional Workshop on Empirical Process Techniques for Dependent Data, to take place November 21-24, 2000 in Copenhagen.

Please make further inquiries to MaPhySto (maphysto@maphysto.dk) or to the organizer Jørgen Hoffmann-Jørgensen (hoff@imf.au.dk).

See also the leaflet containing the programme, abstracts and list of participants:
[ ps-format | pdf-format ]