Seminar on Empirical Processes

117 Pages·1987·2.371 MB·English

by  Peter Gaenssler, Winfried Stute (auth.)

DMVSeminar
Band9
Springer Basel AG
P. Gaenssler
W. Stute
Seminar on Empirical
Processes
1987 Springer Basel AG
Authors
Peter Gaenssler Winfried Stute
Mathematisches lnstitut Mathematisches Institut
Universitiit Miinchen Universitiit Giessen
Theresienstrasse 39 Amdstrasse 2
D-8000 Miinchen 2 D-6300 Giessen
The seminar was made possible through the support of the
Stiftung Volkswagenwerk.
CIP-Kurztitelaufnahme der Deutschen Bibtiothek
Gaenssler, Peter:
Seminar on Empirical Processes I P. Gaenssler ; W.
Stute. -Basel ; Boston : Birkhiiuser, 1987.
(DMV-Seminar ; Bd. 9)
NE: Stute, Winfried:; Seminar on Empirical Processes
<1985, Dusseldorf>; Deutsche Mathematiker-Vereinigung:
DMV-Seminar
All rights reserved.
No part of this publication may be reproduced, stored in a retrieval system, or
transmitted in any form or by any means, electronic, mechanical, photocopying,
recording or otherwise, without the prior permission of the copyright owner.
© Springer Basel AG 1987
Originally published by Birkhliuser Verlag Basel in 1987
ISBN 978-3-7643-1921-2 ISBN 978-3-0348-6269-1 (eBook)
DOI 10.1007/978-3-0348-6269-1
CONTENTS
I. Foundations 1
II. Local and global structure of empirical processes 11
III. Goodness of fit 17
IV. Conditional empirical processes 40
V. Copula processes 49
VI. Empirical processes for censored data 58
VII. Parameter estimation in smooth empirical processes 75
VIII. Bootstrapping 82
IX. Vapnik-Chervonenkis-theory 87
List of Symbols 110
These notes are based on lectures given in the "Seminar on Empirical
Processes" held at Schloss Mickeln, DUSseldorf, from September 8-13, 1985.
According to the intention of the DMV seminar series, we organized the
material so as to give (as we hope) a fresh approach to empirical processes
thereby elaborating some of the main streams of the theory, being aware that
this is necessarily very subjective.
We are grateful to the DUsseldorf people for their assistance and
hospitality during this wonderful week. Many thanks also to the participants
of the seminar for all their interest and the lively discussions.
Finally, we heartily thank Mrs. Lenk (Giessen), who was responsible for
the excellent layout of the present booklet.
P. Gaenssler, W. Stute
1
I. Foup4ations.
Throughout this text we shall assume that t1.t2, ••• ,tn'''' is a finite or
infinite sequence of independent identically distributed (i.i.d.) random
elements in some sample space~ , defined on some probability space (a,A,P).
In most cases~ will be an Euclidean space Rk. For such t•s, write
k
F(t) = P(t~t) , t E R ,
for the corresponding (unknown) distribution function (d.f.). The best studied
non-parametric estimate of F is, for sample size n eft , the empirical d.f.
In other words, Fn is the d.f. of the empirical measure
n
-1~
~n = n ~ot.
i=l 1
where ox is the Dirac-measure in x e ~ •
The theory of empirical d.f.•s is best elaborated fork= 1, i.e. for real
data. In this Foundation section we shall recall some basic facts about
univariate empirical d.f.•s. See Gaenssler apd Stute (1979) and Shorack and
Wellner (1986) for more details.
Roughly speaking, Fn may be viewed either as
(a) a discrete random measure
or as
2
(b) a stochastic process with nondecreasing paths taking their values in
the space D[-~.~] of all right-continuous left-hand limits functions
on [-~.~] (cf. Pollard (1984)).
Viewed as a measure we may compute integrals w.r.t.F ~e.g., for a given score
n
function 1',
n
J1'(t)Fn(dt) = n-1~1'(ti)
i=1
For integrable 1', f1'(t)Fn(dt) ~ f1'(t)F(dt) with probability one. Integrating a
function 1' = 1'(x, y) of two variates, say, w. r. t. the product of F with
n
itself, we get
J
1'(x,y)F (dx)F (dy) = n-2 ~ 1'(t .• t.)
n n ~ 1 J
sn
1~i, j
which is closely related to a U-statistic. Under (b) we may investigate
inverses of Fn
inflt:Fn(t)~ul, 0 < u < 1
For u = i/n, F~1Ci/n) = ti:n' the i-th order statistic of the sample. On the
other hand, evaluating F at the data, we obtain ranks:
n
In summary, we see that many basic statistics may be easily expressed in terms
of F
n
From elementary probability theory there are lots of finite sample and
asymptotic results which apply to empirical distribution functions. First of
3
all
(1.1) F(t)
i.e. Fn(t) is an unbiased estimator for F(t). Furthermore,
(1.2) Var(F (t)) = n-1F(t) (1-F(t))
n
which implies Fn (t) -+ F(t) in quadratic mean. By the strong law of large
numbers
(1.3) F(t) with probability one.
The Glivenko-Cantelli theorem asserts that even uniform convergence holds.
Finally, the CLT yields
a (t) n112[Fn (t)-F(t)) -+ N[O,F(t) (1-F(t))]
n
in distribution. The collection of random variables lan(t):tERI is the
empirical process of sample size n ~ 1. Donsker•s invariance principle states
that
a -+ B0oF in distribution in the spaceD[-~.~]
n
where B0 is a Brownian Bridge. For finite n, the exact distribution of nFn(t)