Empirical Process Techniques for Dependent Data

377 Pages·2002·31.757 MB·English

by  Herold Dehling, Walter Philipp (auth.), Herold Dehling, Thomas Mikosch, Michael Sørensen (eds.)

Empirical Process Techniques
for Dependent Data
Herold Dehling
Thomas Mi kosch
Michael S0rensen
Editors
Springer Science+Business Media, LLC
Herold Dehling Thomas Mikosch
Ruhr-Universităt Bochum University of Copenhagen
Fakultăt fiir Mathematik Laboratory of Actuarial Mathematics
44780 Bochum DK-2100 Copenhagen
Germany Denmark
Michael S0rensen
University of Copenhagen
Department of StatistÎcs and
Operations Research
Copenhagen
DK-2100 Denmark
Library of Congress Cataloging-in-Publication Data
Empirical process techniques for dependent data / Herold Dehling, Thomas Mikosch,
and Michael S0rensen, editors.
p.cm.
Includes bibliographical references.
ISBN 978-1-4612-6611-2 ISBN 978-1-4612-0099-4 (eBook)
DOI 10.1007/978-1-4612-0099-4
1. Nonparametric statistics. 2. Estimation theory. 3. Lirnit theorems (Probability
theory) 1. Dehling, Herold. II. Mikosch, Thomas. III. S0rensen, Michael.
QA278.8.E47 2002
519.5-dc21 2002071106
CIP
AMS Subject Classifications: Primary: 60FXX, 60GXX, 60JXX, 62GXX, 62MXX
© 2002 Springer Science+Business Media New York
Originally published by Birkhăuser Boston in 2002
Softcover reprint of the hardcover 1s t edition 2002
AII rights reserved. This work may not be translated or copied in whole or in part without the written permission of
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ISBN 978-1-4612-6611-2 SPIN 10772146
Reformatted from editors' files by John Spiegelman, Philadelphia, PA.
9 8 7 6 5 4 3 2 1
Contents
Preface ............................................................................ vii
I. A Thtorial on Empirical Process Techniqnes for Dependent Data 1
Empirical Process Techniques for Dependent Data
Herold Dehling and Walter Philipp .................................................. 3
II. Techniques for the Empirical Process of Stationary Sequences 115
Weak Dependence: Models and Applications
Patrick Ango Nze and Paul Doukhan .............................................. 117
Maximal Inequalities and Empirical Central Limit Theorems
Jerome Dedecker and Sana Louhichi .............................................. 137
On Hoeffding's Inequality for Dependent Random Variables
SaraA. van de Geer ............................................................. 161
On the Coupling of Dependent Random Variables and Applications
Florence Merlevede and Magda Peligrad .......................................... 171
Empirical Processes of Residuals
Istwin Berkes and Lajos Horvath ................................................. 195
III. The Empirical Process of Long Range Dependent Processes 211
Asymptotic Expansion of the Empirical Process of Long Memory Moving Averages
Hira L. Koul and Donatas Surgailis ............................................... 213
The Reduction Principle for the Empirical Process of a Long Memory Linear Process
Liudas Giraitis and Donatas Surgailis ............................................. 241
Distributional Limit Theorems for Empirical Processes Generated by Functions
of a Stationary Gaussian Process
Miguel A. Arcones ............................................................... 257
IV. Empirical Spectral Process Techniques 273
Empirical Spectral Processes and Nonparametric Maximum Likelihood Estimation
for Time Series
Rainer Dahlhaus and Wolfgang Polonik ........................................... 275
Empirical Processes Techniques for the Spectral Estimation of Fractional Processes
Philippe Soulier ................................................................. 299
vi Contents
V. The Tail Empirical Process in Extreme Value Theory 323
Tail Empirical Processes Under Mixing Conditions
Holger Drees ................................................................... 325
VI. Bootstrap Techniques 343
On the Bootstrap and Empirical Processes for Dependent Sequences
Dragan Radulovic ............................................................... 345
Frequency Domain Bootstrap for Time Series
Efstathios Paparoditis ........................................................... 365
Preface
This volume aims to bring together empirical process theory with different concepts of depen
dence in probability theory and statistics. This convergence is needed to respond to situations
in which increasingly complicated stochastic models are being used to capture the dependence
structure of real-life data.
Data with complex dependencies are found in fields as diverse as geology, genomics, envi
ronmental sciences, finance, insurance, meteorology, physiology, and telecommunications, to
name several. Present-day computers, with their high speed and large memory requirements,
have made it possible to use models for dependence in statistical applications. Such advances in
technology have brought the inference for stochastic processes into mainstream statistics.
As a matter of fact, the investigation of the properties of estimators and test statistics for
stochastic processes is much more difficult than for models that assume observations to be
independent. Here, another important development in statistical science comes in. Over the last
few decades, it has become evident that empirical process techniques are extremely useful when
it comes to studying the asymptotic properties of parametric as well as nonparametric statistical
procedures. Given the large increase in the number of statistical problems that utilize concepts and
methods from empirical process theory, it is not surprising that a theory of empirical processes
for dependent data has been developed in recent years.
Our main goal is to give an introduction to empirical process techniques for dependent data and
to survey recent developments that are widely scattered over the statistics and probability litera
ture. It is worthwhile stressing that we do not follow standard patterns by sticking to the standard
empirical process for stationary sequences. In our own research we have often experienced the
fact that researchers with quite different interests and expertise have borrowed techniques from
classical empirical process theory without being aware of parallel developments in other areas.
Such areas include spectral analysis of time series, bootstrap for dependent sequences, extreme
value theory for dependent data, and empirical processes for mixing dependent observations,
including processes with long-range dependence. This volume is a unique collection of mate
rial from the diverse areas mentioned. The interested reader will easily find the common spirit
throughout the various parts of this book and, hopefully, will be able to discover analogies with
techniques used in other fields that are not covered in this text.
Earlier monographs on empirical processes such as Empirical Processes with Applications to
Statistics by Shorack and Wellner, Weak Convergence and Empirical Processes by van der Vaart
and Wellner, and Convergence of Stochastic Processes by Pollard cover the case of independent
observations. We believe that this volume is a timely and indispensable supplement to these books.
It can also be seen as a follow-up to the 1986 Birkhauser volume Dependence in Probability and
Statistics, edited by Eberlein and Taqqu, which served as a standard reference for a generation
of researchers.
In Part I, Herold Dehling and Walter Philipp give a tutorial on empirical process techniques
for dependent data. They present a gentle introduction to empirical processes and dependence
from the early roots of Hermann Weyl up to recent developments via the classical contributions
Vlll Preface
of Patrick Billingsley. Dehling and Philipp start with the basics of empirical process theory for iid
data, including the Glivenko-Cantelli theorem and the Donsker empirical central limit theorem,
after which they turn to weakly dependent sequences and sketch the technical arguments needed
in the case of weak dependence. Different mixing properties are explained in detail and illustrated
through various examples. Extensive overviews of covariance inequalities and their applications
to moment bounds for sums of dependent variables are also given. All these tools, in conjunction
with chaining arguments, are then used to derive a multitude of empirical central limit theorems
for mixing sequences.
Dehling and Philipp also present a unique survey of recent developments on the empiricial
process of U -statistics structure. One section is devoted to the limit theory for the empirical
process of strongly dependent or long-memory stationary sequences. An accessible introduction
is given to this field, stressing the crucial differences between the weakly and strongly dependent
worlds of processes. This tutorial chapter, with a bibliography of some 150 references, allows
the reader to obtain a thorough understanding of the basic theory of empirical processes and
dependence as well as its historical development.
Techniques for the empirical process of stationary sequences are presented in Part II. Vari
ous authors give accounts of their research on the classical empirical process in the extended
framework of dependent data. Patrick Ango Nze and Paul Doukhan introduce a new concept
of weak dependence based on the covariances of functions of the stationary sequence. They
apply this notion to derive weak limit theory for dependent sequences, including the empirical
central limit theorem. Jerome Dedecker and Sana Louhichi, in their presentation of a recent
development concerning empirical central limit theorems, examine maximal inequalities which
are crucial for proving tightness of the empirical process for mixing sequences. They work in
the abstract context of the empirical process indexed by classes of functions, assuming entropy
type conditions.
Sara van de Geer continues with a generalization of Hoeffding's inequality to the case of
dependent variables and illustrates the theory with an application to a general nonlinear au
toregressive process. Florence Merlevede and Magda Peligrad continue with the coupling of
sequences of dependent, respectively independent, variables with the same marginals. They ap
ply these results to obtain uniform strong laws of large numbers and central limit theorems
for various classes of dependent sequences. Istvan Berkes and Lajos Horvath review results on
the behavior of empirical processes of residuals from ARMA, ARCH and GARCH time-series
models that are widely used in practice, including financial econometrics.
The empirical process of long-range dependent data exhibits asymptotic behaviour which is
very much different from weakly dependent data. Such empirical processes are treated in Part III,
which is written by masters of their trade and accessible to the nonspecialist who cannot follow
the branching specialized literature.
Hira L. Koul and Donatas Surgailis consider linear processes with long-range dependence
which constitute a major class of time-series models that are important for statistical applications.
They review asymptotic expansions for the empirical process of finite and infinite variance
sequences with long-range dependence and discuss applications to goodness-of-fit testing for
the marginal stationary error distribution in linear regression models and to M -estimation in the
one sample location model.
Liudas Giraitis and Donatas Surgailis continue with an extension of the so-called reduction
principle of Dehling and Taqqu (1989) which is closely related to the asymptotic expansions
of the empirical process of long-range dependent stationary sequences. The latter principle is
a fundamental result which allows one to derive the limiting structure of the empirical process
for long-range dependent processes. The paper by Miguel A. Arcones provides a useful review
of the literature on limit theorems for sums of nonlinear functions of a stationary sequence of
Preface IX
Gaussian random variables with particular emphasis on empirical processes.
In the classical work of Grenander, Rosenblatt, Bartlett and others from the 1950s it was
realized that the integrated periodogram process, used in the spectral analysis of time series for
goodness-of-fit tests, model fitting and change point analysis, can be considered as a quantity with
properties analogous to the classical empirical process. These analogies motivated the adaptation
of ideas and techniques from classical empirical process theory to the spectral analysis of time
series. They are the goal of Part IV on empirical spectral process techniques.
The paper of Rainer Dahlhaus and Wolfgang Polonik makes the links between empirical
process theory and spectral analysis transparent. They show in various examples how empirical
process ideas come naturally into the picture by considering the integrated periodogram process
indexed by appropriate classes of functions. Under entropy conditions the latter process weakly
converges to the corresponding spectral distribution. This fact opens the door to goodness-of
fit tests in the spectral domain, nonparametric Whittle likelihood estimation for time series,
change point analysis and many other statistical topics. They indicate how the theory has to
be modified for the Gaussian and non-Gaussian stationary cases, and for locally nonstationary
processes. They also show that the techniques used are similar to those in empirical process
theory, including exponential inequalities, chaining arguments and others which originate from
the latter theory.
The paper by Philippe Soulier stresses the analogies between the periodogram at the Fourier
frequencies and a sequence of independent identically distributed random variables. Soulier
points out how these similarities can be used to derive limit theory for functionals of the pe
riodogram process. He obtains a functional central limit theorem for the empirical process of
the periodogram process at the Fourier frequencies and shows how these results can be used
in statistical applications to the estimation of the one-step-prediction error of a time series and
goodness-of-fit testing. He also studies the behavior of the periodogram for fractional (long
range dependent) time series whose behavior is totally different from the weakly dependent
case. Again, empirical process techniques playa major role for deriving functional central limit
theorems and the resulting statistical applications.
Part V is devoted to the tail empirical process. The latter is perhaps less familiar; only recently
it has become popular in the extreme value statistics community. The tail empirical process turns
out to be a useful tool for parameter estimation for extremes of dependent data as well as for
tail and high quantile estimation. Holger Drees gives a unique review on the limit theory for tail
processes. It includes Rootzen's result for the uniform tail empirical process and generalizations
thereof, and results for the uniform tail quantile process. The results are applied to derive the
asymptotic normality of statistical tail functionals used to deduce asymptotic normality of various
estimators in extreme value statistics under mixing conditions.
Over the last 15 years bootstrap techniques have been developed for dependent data as well.
Since the bootstrap is based on the empirical distribution of the data, the link with the main
ideas of this volume is quite natural. Therefore, Part VI is entirely devoted to the bootstrap for
dependent data. Dragan Radulovic gives an introduction to the basic ideas in this field and also
surveys recent developments on the bootstrap for stationary sequences, including new results
for Markov chains. Whereas he focuses on the bootstrap based on the empirical distribution
of the sample, Efstathios Paparoditis considers nonparametric res amp ling methods based on
the empirical distribution of the periodogram process. Both papers give a unique overview of
bootstrap techniques for dependent data which are available in the literature.
This volume grew out of a MaPhySto Instructional Workshop on Empirical Process Techniques
for Dependent Data organized by Thomas Mikosch, Spren Feodor Nielsen and Michael Sprensen
in November 2000 at the University of Copenhagen. The workshop was made possible by
generous support from the Centre for Mathematical Physics and Stochastics (MaPhySto) which
x Preface
is funded by the Danish National Research Foundation. Supplementary support was received
from the research training network on Statistical Methods for Dynamical Stochastic Models
(DYNSTOCH) under the EU programme Improving Human Potential and from the Centre for
Analytical Finance (CAF) funded by the Danish Social Science Research Council. Many of the
contributors to this volume taught at the instructional workshop. In particular, a main series of
lectures was given by Herold Dehling and Walter Philip, which becames the starting point for this
volume. We are delighted that the authors here, who are top specialists in their field of expertise,
agreed to contribute to this book.
We take pleasure in thanking Daniel Straumann from the University of Copenhagen, who
masterfully handled the difficult job of merging the different Jb.T}jX styles of the authors. We
would also like to thank Ann Kostant from Birkhauser Boston who carefully supervised the
editing of this volume.
We are convinced that this volume is a useful collection of material on dependent sequences
and empirical processes. The mix of time-series analysis, statistics, empirical process theory,
extreme value theory, and many other areas may be unusual. On the other hand, we feel that
this book will contribute to a better understanding of the parallel developments in different
areas and lead to a beneficial exchange of ideas and concepts. Researchers in statistics, time
series analysis, extreme value theory, probability theory will find a useful toolbox in this book.
The nonspecialized reader or the interested graduate student will get access to the worlds of
dependence and empirical measures.
Herold G. Dehling Bochum, 1 June, 2002
Thomas Mikosch Copenhagen, 1 June, 2002
Michael Sf/Jrensen Copenhagen, 1 June, 2002
Empirical Process Techniques
for Dependent Data