Unimodality of Probability Measures
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 382
Unimodality of
Probability Measures
by
Emile M. J. Bertint
loan Cuculescu
Departement de MatMmatiques et de Statistique,
Universite Laval,
Quebec, Canada
and
Radu Theodorescu
Faculty of Mathematics,
University ofBucharest,
Bucharest, Romania
Springer-Science+Business Media, B. V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4769-4
ISBN 978-94-015-8808-9 (eBook)
DOI 10.1007/978-94-015-8808-9
Printed on acid-free paper
All Rights Reserved
@1997 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1997.
Softcover reprint of the hardcover 1st edition 1997
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
To the memory of our parents, to our families
Contents
xi
Preface
1 Prelude
1.1 Notations
1.2 Radon measures and strict topology .
1.3 Convexity and boundaries . .
1.4 Transforms and convolutions ..
1.5 Miscellany . . . . . . . . . . . .
1.5.1 Quasi concave functions
1.5.2 Convex functions
1.5.3 Correspondences
2 Khinchin structures
2.1 Representing measures
2.2 Choquet representation.
2.3 Khinchin spaces . . . . .
2.4 Khinchin morphisms . .
1
1
3
8
11
14
14
16
16
19
20
26
30
34
2.5
Standard Khinchin spaces
39
2.6
2.7
2.8
Other forms of the Theorem of Khinchin
Khinchin structures on groups
Comments.......
44
3 Concepts of unimodality
3.1 Beta unimodality . . . . . . . . . . . . . . . .
3.1.1 Construction of the Khinchin space ..
3.1.2 Characterizations of beta unimodality .
3.1.3 Further properties of beta unimodality
3.1.4 (ex, 1)- and (1, v)-unimodality
3.1.5 Examples . . . . . . . . . .
3.2 Block Beta unimodality . . . . . .
3.3 Some more concepts of unimodality
3.3.1 Central convex unimodality
3.3.2 Monotone unimodality
3.3.3 Linear unimodality
3.3.4 Schur unimodality ..
Vll
47
52
55
55
56
62
65
70
72
74
75
77
87
87
88
CONTENTS
viii
3.3.5
3.4
3.5
Closed convex sets of star unimodal probability
measures . . . . . . . . . . . . . . . .
Simulation of Khinchin probability measures
Comments...................
· 95
· 104
· 108
4 Khinchin's classical unimodality
4.1 Single-humped probability density functions
4.1.1 Characterization property . . . . .
4.1.2 Iteratively single-humped functions
4.1.3 Maximum likelihood estimators
4.2 Concentration functions . . . . .
4.2.1 Characterization property ..
4.2.2 A representation theorem ..
4.2.3 Location, dispersion, skewness
4.3 Preserving unimodality by mixing .
4.4 Comments...............
·
·
·
·
·
·
·
·
·
·
111
111
112
114
118
123
125
128
133
136
138
5 Discrete unimodality
5.1 Unimodality on the set of all integers
5.1.1 Several definitions. . . . . . .
5.1.2 The mean-median-mode inequality
5.1.3 Variance upper and lower bounds
5.1.4 Mixing discrete distributions . . . .
5.1.5 Concentration functions . . . . . .
5.2 A one-parameter class of random variables
5.3 A two-parameter class of random variables
5.3.1 Preliminaries . . .
5.3.2 Basic properties ..
5.3.3 Further properties
5.4 Comments.........
·
·
·
·
·
·
·
·
·
·
·
·
143
143
143
148
151
154
159
166
168
169
171
175
179
6 Strong unimodality
6.1 Strong unimodality, logconcavity, and dispersivity
6.2 Multiplicative strong unimodality
6.3 Discrete strong unimodality
6.4 Comments............
183
· 183
· 190
· 198
· 199
7 Positivity of functional moments
7.1 Problem 234 . . . . . . . . . . .
7.2 Mean preserving representations.
7.2.1 General representations ..
7.2.2 Specific representations ..
7.2.3 Characterization property
7.3 Slantedness . . . . . . . . . . . .
7.3.1 Main tools . . . . . . . . .
7.3.2 Conditions for slantedness
201
· 201
· 202
.203
· 206
· 211
· 215
· 215
.217
CONTENTS
7.4
7.3.3 Signed moments. . . . . . . . . .
7.3.4 About the concept of slantedness
Comments . . . . . . . . . . . . . . . . .
IX
. 220
.220
.222
Bibliography
225
Symbol index
241
Name index
243
Subject index
247
Preface
Labor omnia vincit improbus.
VIRGIL, Georgica I, 144-145.
In the first part of his Theoria combinationis observationum erroribus minimis obnoxiae, published in 1821, Carl Friedrich Gauss [Gau80, p.10] deduces a
Chebyshev-type inequality for a probability density function,
when it only has the property that its value always decreases, or at least does
not increase, if the absolute value of x increases l .
One may therefore conjecture that Gauss is one of the first scientists to use the
property of 'single-humpedness' of a probability density function in a meaningful
probabilistic context.
More than seventy years later, zoologist W.F.R. Weldon was faced with 'doublehumpedness'. Indeed, discussing peculiarities of a population of Naples crabs, possibly connected to natural selection, he writes to Karl Pearson (E.S. Pearson [Pea78,
p.328]):
Out of the mouths of babes and sucklings hath He perfected praise! In the
last few evenings I have wrestled with a double humped curve, and have
overthrown it. Enclosed is the diagram... If you scoff at this, I shall never
forgive you.
Not only did Pearson not scoff at this bimodal probability density function, he
examined it and succeeded in decomposing it into two 'single-humped curves' in his
first statistical memoir (Pearson [Pea94]).
Around 1938 Aleksandr Yakovlevich Khinchin [Khi38] revealed in a fundamental
paper the intimate relationship between the set of all single-humped probability
density functions on the real axis, with a given hump, and the set of all probability
measures on R, known today as the Theorem of Khinchin. Thirty years later,
around 1970, it was felt that the Theorem of Khinchin should be considered as a
special, but noncompact, form of the Theorem of Krein-Milman or of the Theorem
of Choquet. This question was answered, in a positive sense, in [BT84a].
The last decade has seen a growing number of publications on unimodality,
generalizing this notion to higher dimensions, to other spaces, or to other types
1
For a German version of the original text written in Latin, see Gauss [Gau64, p.9]
xi
xii
PREFACE
of unimodality. Khinchin-type representation theorems mayor may not hold for
these generalizations. In the affirmative case, the set U of unimodal probability
measures, with a prescribed mode, is a closed convex set for which the conclusion of
the Theorem of Krein-Milman holds: U is the closed convex hull of the set ~ of its
extreme points. The representation theorem then exhibits an isomorphism between
U and the set 'P(~) of all probability measures on ~, and can again be formulated
as a generalization of the Theorem of Choquet-Meyer in convexity theory.
For the past fifteen years or so, Emile and I devoted our efforts to the study
of different concepts of unimodality. This monograph evolved in conjunction with
examining what these concepts have in common. It turned out that one of the basic
features was the representation theorem. This result is like a fine jewel that reveals
its beauty under illumination from varying positions. So, several years ago, Emile
and I started working on a project, setting Khinchin-type representation theorems
as the central theme of such a monograph. It happened to be a fortunate choice.
In 1988 the excellent monograph "Unimodality, Convexity, and Applications" by
Dharmadhikari and Joag-dev [DJ88] appeared, containing a wealth of material on
unimodality. Our approach is different. We first developed an abstract framework
for unimodality, and, as Pal Erdos says, turned a lot of coffee into theorems. This
framework, an example of applied functional analysis, is then used for the introduction of different types of unimodality and the study of their behaviour. We also
provide several useful consequences or ramifications tied to these notions.
This monograph is neither an encyclopedia nor a book on the history of unimodality. Its first aim is to serve as an understanding of the basic features of
unimodality. In a span of less than two years, Emile and I had written about three
quarters of it. In the fall of 1993 we met in Quebec for an intensive working session.
Unfortunately, it was the last one. By Christmas, Emile had become aware that
he was terminally ill. After a short but courageous battle with cancer, he passed
away on March 23, 1994. In spite of the difficulties raised by his sudden death, I
continued the work. In June 1995 I met loan Cuculescu, a former colleague of mine,
who joined me in the effort to finish this monograph, trying to keep as much as
possible to our original project.
Chapter 1 lays out those basics that are needed for the understanding of the
mathematical reasoning in later chapters.
Chapter 2 starts with a study of convex sets which are closed but not compact, situated in particular topological vector spaces, namely in a space of measures
on a completely regular space; the convex sets examined consist of probability measures. Furthermore, Chapter 2 deals consistently with the concept of Khinchin space
which we shall keep always in mind when examining different types of unimodality.
In short, a Khinchin space is a - not necessarily compact - Bauer simplex. We
also introduce the concept of Khinchin morphism between Khinchin spaces, obtaining in this way a category. Next we construct the product of two Khinchin spaces.
Versions of the concept of Khinchin space, as that of standard one, corresponding to
a parametrization of the set of extreme elements, are adapted to, and motivated by,
unimodality. In the last part of the chapter, the structure of Khinchin space leads to
PREFACE
xiii
the Levy-Shepp refinement of the Theorem of Khinchin, describing unimodal probability measures as the result of a certain 'action' on a fixed generating probability
measure.
Building on the results of Chapter 2, Chapter 3 introduces a new notion of un imodality, called beta unimodality, defined on a Hilbert space and generated by the
beta distribution function. Beta unimodality contains most of the existing multivariate notions of unimodality, and in particular univariate classical unimodality, as
special cases. Special attention is also given to block beta unimodality. A discussion on several existing multivariate notions of unimodality concludes this chapter.
In addition to the concept of star unimodality, closely related to beta unimodality, we examine central convex, monotone, linear, and Schur unimodality, and show
that the corresponding sets of probability measures are not simplexes. However, in
the central convex and in the Schur cases, these sets contain simplexes which are
naturally related to the whole sets.
Chapter 4 concerns Khinchin's classical unimodality. Here we insist on the characterization of unimodality in terms of quasi concavity and concentration. We also
examine the problem of preserving unimodality by mixing.
Chapter 5 is devoted to discrete unimodality. After a short discussion on several
concepts of unimodality, we explain the rationale of introducing our concepts of
discrete unimodality. It essentially consists in building Khinchin structures yielding
Choquet-type representations. Such representations are used, for example, in dealing
with the preservation of unimodality by mixing. We also obtain discrete analogues
of certain results in Chapter 4.
Chapter 6 is dedicated to the concept of strong unimodality on R and to Ibragimov-type results characterizing the probability measures which preserve unimodality by - additive or multiplicative - convolution. The relationship to certain
dispersive orders is also indicated. Next the discrete case is briefly discussed.
Chapter 7 deals with the concept of slantedness. A certain representation of
a unimodal probability measure /-L on R or Z as a mixture of simpler probability measures, each of them having the same expectation as /-L, is studied. This
representation and Choquet-type results are then used to prove the positivity of
odd central moments (slantedness, in a more general context).
The best approach to reading this monograph is to start from the beginning,
perhaps temporarily skipping Chapter 1, and to go through each page until the last
one. The second best approach is to start from the beginning and to skip some timeconsuming material without loss of continuity. For example, many 'tricky' proofs
may thus be so omitted.
Every chapter, except the first one, concludes with a section consisting of comments. They refer to historical aspects or provide complementary information and
open questions.
The bibliography covers only entries referred to in the text. It is in no way
exhaustive. Symbol, name, and subject indexes ensure quick and easy access to all
information.
xiv
PREFACE
Theorems, propositions, lemmas, corollaries, definitions, remarks, and notes are
continuously numbered per section, whereas formulas are numbered per chapter.
Theorem 2.6.2 is the second formal statement of Section 2.6 and (2.4) is the fourth
formula of Chapter 2.
The first beneficiary of this book is the researcher confronted with unimodality as
well as the investigator in the many fields of its application. Although the monograph
has not been designed as a textbook, the material could form the basis for a graduate
course or a seminar on unimodality, or, eventually, on applications of convex analysis.
Prefaces tend to end with a list of thank you's, sometimes too long. So, the
authors will merely thank their families for their on-going encouragement and a
continuous supply of strong coffee. The support of the National Sciences Research
Council of Canada, of the Fonds F.C.A.R. of the Province of Quebec, of the Nederlandse Organisatie voor Wetenschappelijk Onderzoek, as well as of the Universiteit
van Utrecht and of the Universitiit der Bundeswehr Hamburg, which have sponsored
in part our project, is acknowledged with appreciation.
I hope that this text, on which Emile made changes until the very last minute,
would have had his approval.
Radu Theodorescu
Quebec, August 1996
Chapter 1
Prelude
One of the main topics of this monograph is the Theorem of Khinchin stating that
every unimodal probability measure on the real line is an integral of uniform probability measures on intervals. This assertion appears as a Choquet-type theorem
concerning representations of points in convex sets as barycenters, the 'points' being
in our case themselves measures.
In general, we do not assume separability conditions. Therefore the main topics
of this chapter are a general framework for measure theory, namely that of Radon
measures on completely regular spaces (Section 1.2), and the Choquet theory (Section 1.3). Material on transforms and convolutions (Section 1.4), and on quasi
concavity, convexity, and correspondences (Section 1.5) is also included.
Standard results are borrowed from the following references: Bourbaki [Bou71]
and [Bou74] for topology, Badrikian [Bad70], Bourbaki [Bou69] or Tjur [Tju80] for
measure theory, Jarchow [Jar81], Bourbaki [Bou81], Cooper [Coo78], Phelps [Phe66]
for functional analysis, and Asimow and Ellis [AE80], Alfsen [Alf71], Meyer [Mey66],
and Rockafellar [Roc70] for convexity theory. Results, less easily found in these
references, are stated as lemmas or propositions.
1.1
Notations
In this section several notations, used throughout this monograph, are listed.
Let E be a topological linear space and let X, Xl, and X 2 be completely regular
topological spaces. Given a function f on X and the functions h on Xl and h on
X 2 , a subset H of an ordered vector space, and a set 1£(X) of numerical functions
on X, we denote by:
1
E. M. J. Bertin et al., Unimodality of Probability Measures
© Springer Science+Business Media Dordrecht 1997
CHAPTER 1. PRELUDE
2
_
0
A,A
AC
8(X)
C(X)
Diam(A)
E'
Ex
f
flA
f(A)
h0h
f+, ff+(x)
f-(x)
H±
1ib(X)
1ic (X)
K(X)
C}(X, JL)
the closure and the interior of A;
the complement of A;
the set of all Borel-measurable numerical functions on X;
the set of all continuous real-valued functions on X;
the diameter of A, with respect to a given metric;
the dual space of E;
the degenerate probability measure at x, defined by EX(f) = f(x)
for f E Cb(X);
the map x H limsuPA3Y-+x f(y), on A, the upper semicontinuous
regularization of f;
the restriction of f to A;
the image of A under the function f;
the map (Xl, X2) H h(xdh(X2) on Xl x X 2;
the positive and the negative part of f
liffiy.j.x f(y), where f : X --+ Rand X C R;
f+( -x);
the set of all positive (negative) elements of H;
the subset of all bounded real-valued functions of 1i(X);
the set of all finite f E 1i(X) with compact support;
the set of all compact subsets of X;
the space of JL-integrable functions on X, JL E M~(X), endowed
with the seminorm II· lit = f H JL(lfl);
Ll(X, JL) the associated Banach space of equivalence classes of JL-integrable
functions;
the set of all Radon measures on X;
M(X)
Mb(X) the set of all bounded Radon measures on X, endowed with the
narrow topology a(Mb(X), Cb(X));
the
exterior measure and the interior measure associated with JL;
ll,~
the
product measure on Xl x X 2 of JLs E Mb(Xs), S = 1,2;
JLl 0 JL2
the set N \ {O} of all strictly positive natural numbers;
N*
the set of all Radon probability measures on X;
P(X)
Pl ®P2 the linear hull of the set of all h 0 12, where fs E Ps c Cb(Xs ) for
s = 1,2;
the image measure of JL E Mb(X) by 7T, where 7T : X --+ Xl is a
7T' (JL)
JL-measurable map;
JL(f)
J f dJL;
the two-point compactification [-00,00] of the set R of all real
R
numbers;
the support of the measure JL E Mb(X);
supp(JL)
a(E,E') the weak topology on E, defined by the dual pairing (E, E');
a(E',E) the weak topology on E', defined by the dual pairing (E', E);
X
the Stone-Cech compactification of X;
the indicator function of A.
1A
The notations JL-a.e. or JL-a.s. denote almost everywhere or almost surely with
respect to the measure JL. For n-dimensional Lebesgue measure, usually denoted by