Asymptotic Distribution Theory in Nonparametric Statistics

212 Pages·1985·6.378 MB·German

by  Manfred Denker (auth.)

Advanced lectures in mathematics
Manfred Denker
Asymptotic distribution
theory in nonparametric
statistics
Manfred Denker
Asymptotic Distribution Theory
in Nonparametrie Statistics
Advanced Ledures
in Mathematics
Edited by Gerd Fischer
Jochen Werner
Optirnization. Theory and Applications
Manfred Denker
Asymptotic Distribution Theory
in Nonparametric Statistics
Manfred Denker
Asymptotic
Distribution Theory in
Nonparametrie Statistics
Springer Fachmedien Wiesbaden GmbH
CIP-Kurztitelaufnahmeder Deutschen Bibliothek
Denker,Manfred:
Asymptoticdistribution theory in nonparametric
statistics1Manfred Denker.- Braunschweig;
·Wiesbaden:Vieweg.1985.
(Advancedlecturesin mathematicsl
ISBN 978-3-528-08905-4 ISBN 978-3-663-14229-4 (eBook)
DOI 10.1007/978-3-663-14229-4
1985
All rights reserved
© SpringerFachmedienWiesbaden 1985
Ursprünglicherschienen beiFriedr.Vieweg& SohnVerlagsgesellschaftmbH, Braunschweig 1985.
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PREEACE
During the last years I gave som~ courses at the University of
Göttingen on selected topics in nonparametric statistics, main
lyon its a3ymptotic distribution theory. They were intended
for advanced students with a good mathematical background,
especially in probability theory and mathematical statistics.
The present notes result from these lectures.
Three basic types of statistics are treated in this book: Chap
ter 1 contains Hoeffding's U-statistics, chapter 2 differenti
aLle statistical functionals and chapter 3 statistics based on
ranks. Although the emphasis lies on the asymptotic distribu
tion theory for these statistics I intended to give some moti
vation from two viewpoints. The examples cover more than is
needed for illustrations and each chapter contains some result
on optimality properties. Chapter 4 on efficiency and conti
guity may be regarded as an application of the results in the
preceding chapters.
The results are presented with complete proofs. For this rea
son a good deal of probability is needed for parts of the
proofs. Billingsley's book (1968) is an adeguate reference
for it. For example, Donsker's theorem on the weak convergence
of the empirical process to the Brownian bridge is the essen
tial point for the development of the asymptotic distribution
theory for von Mises' statistics in chapter 2, section 2. For
most of the results in the other sections, however, a basic
knowledge in probability suffices, like the central l imi t theo
rem under the Lindeberg condition.
I intended to restrict the content to results which can be
proved in a reasonable way during a course, even concerning
its technical part. Thus some of the theorems do not appear
in full generality and very often, the assumptions are not
as weak as possible. Unfortunately, the asymptotic oistribu
tion theory in nonparametric statistics reguires same rather
extensive computations. The technigues for doing this I tried
to develop carefully for each of the statistics under conside
ration. Some of the material in this book has not yet appeared
v
in print elsewhere and therefore I hope that these notes are
helpful not only for students to get acquainted with some theo
retical aspects of nonparametrie asymptotic distribution theory.
Certainly, in a volume like this, I had to make choices of the
material to be included and the reader rnight miss essential
and important other classes of statistics, e.g. Kolmogorov
Smirnov statistics, statistics based on empirical processes in
general or statistics treated by martingale metho0s, to name a
few. In recent years some books were puhlished containing
more and different matters about the mathematical theory in
nonparametrie statistics, on the other hand these notes go
beyond previous presentations.
I am very much indebted to Chr. Grillenberger, G. Keller and
U.Rösler, whose valuable collaborations resulted in joint
papers and made this book possible. Especially, I have to
thank H.Dehling and Chr.Grillenberger for helpful comments
while reading parts of the manuscript, and also M.Powell for
assisting in typing. Also, I am grateful to the Vieweg Verlag
for publishing the manuscript.
Göttingen, September 1984
VI
CONTENTS
Chapter 1: U-statistics 1
1. Definition of U-statistics 1
2. The decomposition theorem for a U-statistic 9
3. Convergence theorems for U-statistics 18
4. Generalized U-statistics 32
Notes on chapter 1 50
Chapter 2: Differentiable statistical functionals 51
1. Definition of differentiable statistical
functionals 51
2. The asymptotic distribution of differentiable
statistical functionals 68
3. M-estimators 89
Notes on chapter 2 96
Chapter 3: Statistics based on ranking methods 98
1. Permutation tests 98
2. Simple linear rank statistics 106
3. A representation of simple linear rank statistics 116
4. Asymptotic normality of simple linear rank
statistics 126
5. Signed rank statistics and R-estimators 138
6. Linear combinations of a function of the order
statistic 159
Notes on chapter 3 168
Chapter 4: Contiguity and efficiency 169
1. Pitman efficiency 169
2. Contiguity of probability measures 187
Notes on chapter 4 198
References 199
Subject index 202
VII
CHArTER 1: U-STATlSTlCS
Hoeffding introduced U-statistics in 1948, partly influenced
by earlier work of Halmos, and closely connected to von ~~ises'
functionals (von Mises, 1947). U-statistics can be viewed as
a class of unbiased estimators of a certain parameter, based
on some averaging procedure.
We shall investigate the asymptotic properties of non-degener
ate U-statistics in the next three sections; section 4 gives
extensions to LehmannIs generalized lJ-statistics. ~1ore results
on U-statistics will appear in the next chapters, especially
the degenerate case in chapter 2 and contiguity in chapter 4.
1. Definition of U-statistics
For a measurable space (E,B) denote by ~1(B) the set of all
probability measures on B •
Definition 1.1.1: A function 8: Mo R, defined on a sub
4
set Ho C M(B), is said to be regular (estimable) w.r.t. ~~o'
if there exist an integer m ~ and an unbiased estimator h
based on m independent, identically distributed (= i.i.d.),
E-valued random variables.
This definition means that there is a measurable map
h : Ern ....IR such that
8(P)
where pm is the m-fold Cartesian product of P If 8 is
given, the minimum of all integers m with this oroperty is
called the rank of 8. It should be remarked that h can
always be assumed to be symmetrie. Indeed, if Y denotes
m
the permutation group acting on Ern by nerMuting the coordi
nates, then (mJ)-l I hoc is an unbiased symmetrie esti-
mator for 8 cEYm (assuming that h is unbiased).
The definition of a U-statistic is based on this notion of
regular functions 8.
Definition 1.1.2: Let :3: r' -+IR be regular. If
o
h : Ern -+ ffi is a symmetric, unbiased estimator for :3, then
for any n ~ m the map
i s called a U-statistic. Here, IT denotes the set of al l
n
n
maps TI : E -+ Ern which arise from projections onto m co
ordinates ~ i < i z < . .. < im ~ n. The estimator h is
l
called the kernel of the U-statistic and m i s called its
degree.
Note that U (h) is again an unbiased, symmetric estimator
n
for :3 based on n LLd. observations Xl' ••• ,X wi t.h
n
distribution P E r~ Then the U-statistic can be written as
0
U (h) (n)-l I h(X. , . . . ,X. );
n m . . l.1 ~m
1$~ I <· ·<lm:>n
for convenience, the dependence on the Xi 's i5 suppressed in
Example 1.1.1: Let f :E-+1R be a measurable function and
11' = {P E H(B) f E L }. Then :3IP) = Jf dP is regular
0 1(P)
(of rank 1) and h = f is an unbiased estimator of rank 1.
k
For f Ix) = x (E = IR), one has the k-th moment and the cor-
I
x~
responding U-statistic l/n is known as the sample
i=l l.
k-th moment (for k=l this l.S the sample mean) .
I f E = R, fIx) = 1(__,t](X) It E R) estimates FIt), where
F denotes the right continuous distribution function (= d.f. )
of Xl ' The corresponding U-statistic is just the empirical
d.f. of X " 'X evaluated at t
1" n
: J. 2
Example 1.1.2: Let E = IR and f1 IP x dP (x) < 00} • In
order to estimate :3(P) Jx2 dP0(x) - (.I. x c1PIx)) 2, the
variance of P , one can use the estimator h(x,y) 1/2 (x_y)2
of rank 2. The corresponding U-statistic gj,ves the sampl e
variance
1 n 2 n 2
I n-l [I x. - (I x.) ).
l~i<jSn i=l 1 j=l J
2