Hi:
I was asked by Ole Barndorff-Nielsen to point out to econometricians a
summer school (August 9-20, 1999) which is being organised at Dept of Maths,
Aarhus University in Denmark on Empirical Processes. Given below is a more
detailed description of the summer school. Contact details are given towards
the end of the text.
Neil Shephard
MaPhySto
Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus
Funded by The Danish National Research Foundation
MaPhySto Summer school on
Empirical Processes
From Monday, August 9, 1999 to Friday, August 20, 1999 MaPhySto will
organize a summer school on Empirical
Processes. Each of the following have agreed to give a series of
lectures (7-9 lectures of 45 min.):
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 will be based on parts of a book by the
author, also called Uniform Central Limit
Theorems to be published by Cambridge University Press,
probably in the first half of 1999.
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-J{\o}rgensen (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.
Furthermore we intend to have 5-10 guest speakers (still leaving
slots for the participants to speak).
The Summer School will take place in Aarhus at the Department of
Mathematical Sciences, University of Aarhus.
Support
Limited support is available. Preference will be given to younger
participants.
Registration
Please register via the registration form as soon as possible before
June 1, 1999. The registration fee is DKK 400
(approx. $60).
More Information
Please make further inquiries to MaPhySto ([log in to unmask]) or to
the organizer Jørgen Hoffmann-Jørgensen
([log in to unmask]).
This document, http://www.maphysto.dk/events/EmpirProc99/index.html,
was last modified September 14, 1998.
--------------------------------------------------------
Dr. Neil Shephard
www.nuff.ox.ac.uk/economics/people/shephard.htm
Office Home
Phone: 44 1865 278593 44 171 286 2476
Fax:44 1865 278621
Nuffield College, 43 Delaware Mansions
Oxford OX1 1NF, UK. Delaware Rd
London W9 2LH
and
(part-time) 15 July - 15 December 1998
Isaac Newton Institute, Cambridge University
20 Clarkson Rd
Cambridge CB3 0EH, UK
Phone: 44 1223 330540
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