Table Of ContentDistributions with given Marginals and Moment Problems
Distributions with
given Marginals and
Moment Problems
Edited by
Viktor BeneS
Department of Mathematics,
Czech Technical University,
Prague, Czech Republic
and
Josef Stepän
Department of Statistics,
Charles University,
Prague, Czech Republic
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A CLP. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-6329-6 ISBN 978-94-011-5532-8 (eBook)
DOI 10.1007/978-94-011-5532-8
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.
TABLE OF CONTENTS
PREFACE vii
ACKNOWLEDGEMENT x
G. A. ANASTASSIOU / Optimal bounds on the average of a rounded
off observation in the presence of a single moment condition 1
T. RYCHLIK / The complete solution of a rounding problem under two
moment conditions 15
V. GIRARDIN / Methods of realization of moment problems with en-
tropy maximization 21
E. KAARIK / Matrices of higher moments: some problems of represen-
tation 27
L. B. KLEBANOV, S. T. RACHEV / The method of moments in to-
mography and quantum mechanics 35
V. BENES / Moment problems in stochastic geometry 53
M. SCARSINI, M. SHAKED / Frechet classes and nonmonotone de-
pendence 59
A. CHATEAUNEUF, M. COHEN, 1. MEILIJSON / Comonotonicity,
rank-dependent utilities and a search problem 73
P. CAPERAA, A. L. FOUGERES, C. GENEST / A stochastic ordering
based on a decomposition of Kendall's tau 81
M. J. W. JANSEN / Maximum entropy distributions with prescribed
marginals and normal score correlations 87
. V. E. PIPERIGOU / On bivariate distributions with Polya-Aeppli or
Li iders-Delaporte marginals 93
E.-M. TIlT, H.-L. HELEMAE / Boundary distributions with fixed mar-
~~ ~
X. LI, P. MIKUSINSKI, H. SHERWOOD, M. D. TAYLOR / On ap-
proximations of copulas 107
G. DALL'AGLIO / Joint distributions of two uniform random variables
when the sum and difference are independent 117
R. B. NELSEN, G. A. FREDRICKS / Diagonal copulas 121
G. A. FREDRICKS, R. B. NELSEN / Copulas constructed from diag-
onal sections 129
C. M. CUADRAS, J. FORTIANA / Continuous scaling on a bivariate
copula 137
H. G. KELLERER / Representation of Markov kernels by random map-
pings under order conditions 143
J. STEP AN / How to construct a two dimensional random vector with
a given conditional structure 161
A. HIRSHBERG, R. M. SHORTT / Strassen's theorem for group-valued
charges 173
vi
G. LETAC / The Lancaster's probabilities on R2 and their extreme
points 179
M. STUDENY / On marginalization, collapsibility and precollapsibili-
ty 191
J. DUPACOV A / Moment bounds for stochastic programs in particular
for recourse problems 199
T. SZANTAI / Probabilistic constrained programming and distributions
with given marginals 205
V. KANKOV A / On an £-solution of minimax problem in stochastic
programming 211
T. ViSEK / Bounds for stochastic programs - nonconvex case 217
R. JIROUSEK / Artificial intelligence, the marginal problem and in-
consistency 223
O. KIlii / Inconsistent marginal problem on finite sets 235
V. L. LEVIN / Topics in the duality for mass transfer problems 243
C. S. SMITH, M. KNOTT / Generalising monotonicity 253
L. RUSCHENDORF, 1. UCKELMANN / On optimal multivariate cou-
plings 261
L. UCKELMANN / Optimal couplings between one-dimensional distri-
butions 275
D. RAMACHANDRAN, 1. RUSCHENDORF / Duality theorems for
assignments with upper bounds 283
J. H. B. KEMPERMAN / Bounding the moments of an order statistics
if each k-tuple is independent 291
SUBJECT INDEX 305
PREFACE
The last decade has seen a remarkable development of the "Marginal
and Moment Problems" as a research area in Probability and Statistics.
Its attractiveness stemmed from its lasting ability to provide a researcher
with difficult theoretical problems that have direct consequences for appli
cations outside of mathematics. The relevant research aims centered mainly
along the following lines that very frequently met each other to provide sur
prizing and useful results :
-To construct a probability distribution (to prove its existence, at least)
with a given support and with some additional inner stochastic property
defined typically either by moments or by marginal distributions.
-To study the geometrical and topological structure of the set of prob
ability distributions generated by such a property mostly with the aim to
propose a procedure that would result in a stochastic model with some
optimal properties within the set of probability distributions.
These research aims characterize also, though only very generally, the
scientific program of the 1996 conference "Distributions with given
marginals and moment problems" held at the beginning of September
in Prague, Czech Republic, to perpetuate the tradition and achievements of
the closely related 1990 Roma symposium "On Frechet Classes" 1 and
1993 Seattle"A MS Summer Conference on Marginal Problem".
A careful selection of the contributions presented to the 1996 conference
is published in these proceedings. It has been an aim of the editors not only
to reflect the recent developments in the field, but also, with an indispens
able assistance of numerous referees, to publish new and important results.
These are the reasons that make the editors believe that the book will prove
itself useful even for the probabilists and statisticians working outside its
specialized domain and will surely stimulate the research in the "Moments
& Marginals" area itself.
1 For the proceedings see "Advances in Probability Distributions with Given
Ma.rginals". Kluwer Academic Publishers, Dodrecht 1991.
vii
viii
Roughly speaking, the contributions of the proceedings belong to one of
the following topics centered around the invited papers written be carefully
selected scholars :
- Moment Problems and Applications. G.A.Anastassiou applies
Geometric Moment Theory to a statistical problem over observations sub
jected to one moment condition. L.B.Klebanov and S. T.Rachev introduce
Computer Tomography and Quantum Mechanics as research fields where
the moment and marginals technologies may be applied to get new intrigu
ing results - a solution of the computer tomography paradox, for example.
J.H.B.Kemperman shows how to calculate lower and upper bounds of the
moments of an order statistics if each k-tuple in the sample is formed be
independent random variables with a known probability distribution.
- Marginal Problems and Stochastic Order. M.Scarsini and M.Sha
ked examine in their paper non monotone dependence bivariate stochastic
models in the framework of a Frechet class. By nonmonotone dependence is
meant the tendency of a bivariate distribution to concentrate its probability
mass around the graph of a (not necessarily monotone) function.
- Copulas. This well established topic with a remarkable history in
the field is represented by "On Approximation of Copulas" by X.Li, P.Mi
kusinski, H.Sherwood, and M.D. Taylor. The authors observe that uniform
convergence of copulas is not completely satisfactory when trying to verify
some of their properties that could be derived first on a suitable class of
"simple" copulas and then extended to all copulas by a limiting process.
Two different ways of approximating copulas are proposed (Checkerboard
and Bernstein approximations) to meet the needs mentioned above.
- Marginal and Moment Problems - Measure Theory Approach.
G.Letac comes back to a problem with the long history namely that of find
ing extremal measures in the convex set of probability measures with a fixed
pair of marginals. The problem is attacked, this time, in the framework of
Lanchester R2-probabilities. H.G.Kellerer treats the problem of represent
ing a Markov kernel that connects a pair of ordered spaces by an order
preserving random map. The problem is a natural counterpart to the pre
vious studies on Markov kernels represented by continuous random maps.
- Stochastic Programming and Artificial Intelligence. J.Dupa
cova discusses in her paper different methods for bounding the optimal
values of stochastic programs, in particular for recourse problems. R.Jirou
sek studies the problem of constructing a distribution with given marginals
as an inherent part of artificial intelligence approaches to applied problems.
In this context, only the constructions that can be handled efficiently by
computer programs are acceptable.
- Optimizations in Marginal Problem The paper by L.Levin con
cerns the duality theory for nontopological versions of mass transport prob-
ix
lem with the aim to extend the classical Monge-Kantorovich results. L.Rii
schendorf and 1. Uckelmann treat optimal multivariate problems, prove the
existence of optimal Monge solutions in this context and propose an ex
plicit construction method for the optimal transportation in the case that
one mass distribution is discrete. Finally, D.Ramachandran and L.Riischen
dorfformulate a version of the nonatomic assignment .problem with upper
admissible assignments in the form of a dominating measure and obtain a
general duality theorem for this model.
The invited papers, the achievements of which have been reviewed above
are accompanied by 25 contributed papers that, according to the editors'
opinion, increase significantly the scientific value of the proceedings. Cordial
thanks are due to all authors and referees.
Prague, January 1997
Josef Stepan
ACKNOWLEDGEMENTS
The editors desire in the first place to acknowledge that it would never
have been possible to organize the 1996 Prague Conference "Distributions
with Given Marginals and Moment Problems" and to prepare its Proceed
ings for the publication if it had not been for the research grant they have
received from the Grant Agency of the Czech Republic in 1994 under
the contract 201/94/0471. The grant not only encouraged their own sci
entific work but also provided an invaluable help in keeping and promoting
necessary international contacts.
The editors are no less indebted to Faculty of Mathematics and Physics
of Charles University in Prague, Faculty of Mechanical Engineering of
Czech Technical University in Prague, Institute of Information Theory and
Automation in Academy of Sciences of Czech Republic and to Technical
Faculty of Czech Agricultural University in Prague which institutions spon
sored generously the Conference activities.
Among many individuals to whom the organizers of the Conference and
the editors of its Proceedings owe for advice, help and encouragement are
firstly the members of the Conference scientific and organizing committees:
L. Riischendorf (Freiburg i. Br.), H. Sherwood (Orlando), S. T. Rachev
(Santa Barbara), J. Dupacova, V. Slavik, M. Studeny, P. Lachout, D. Hlu
binka, T. Visek.
D. Hlubinka and T. Visek also have the editor's grateful thanks for
preparing the manuscript technically. Last but not least cordial thanks are
due to H. Jandova for the skill and patience with the editors when mastering
their extensive correspondence and paperwork.
Viktor Benes and Josef Stepan
x
OPTIMAL BOUNDS ON THE AVERAGE OF A ROUNDED
OFF OBSERVATION IN THE PRESENCE OF A SINGLE MOMENT
CONDITION
GEORGE A. ANASTASSIOU
Department of Mathematical Sciences
The University of Memphis
Memphis, TN 38152 U.S.A.
Abstract. The moment problem of finding the maximum and minimum
expected values of the averages of nonnegative random variables over
various sets of observations subject to one simple moment condition
is encountered. The solution is given by means of geometric moment
theory (see [3]).
1. Background
r'l
Here [.j and stand for the integral part (floor) and ceiling of the
number, respectively. °
We consider probability measures p, on A := [0, aj, a > or [0, +00).
We would like to find
Ua := sup JfA ([t] + rtl) p,(dt) (1)
2
p.
and
La := inf f ([tj + rtl) J.L(dt) (2)
JA
2
p.
subject to L
tT J.L( dt) = dT, r > 0, (3)
°
where d > (here the subscript in Ua, La stands for average).
Q
V. Bend and J. Stepan (ew.). Distributions with given Marginals and Momenl Problems. 1-13.
@ 1997 Kluwer Academic Publishers.
