Mathematical Statistics and Probability Theory: Proceedings, Sixth International Conference, Wisła (Poland), 1978

395 Pages·1980·12.895 MB·English

by  Odd Aalen (auth.), Dr. Witold Klonecki, Dr. Andrzej Kozek, Dr. Jan Rosiński (eds.)

Lecture Notes in Statistics
Vol. 1: R. A. Fisher: An Appreciation. Edited by S. E. Fienberg and D. V. Hinkley.
xi, 208 pages, 1980.
Vol. 2: Mathematical Statistics and Probability Theory. Proceedings 1978. Edited.
by W. Klonecki, A. Kozek, and J. Rosinski. xxiv, 373, 1980.
Springer Series in Statistics
L. A. Goodman and W. H. Kruskal, Measures of Association for Cross Classi
fications. x, 146 pag~s, 1979.
J. O. Berger, Statistical Decision Theory: Foundations, Concepts, and Methods.
xiv, 420 pages, 1980.
Lecture Notes in
Statistics
Edited by S. Fienberg, J. Gani, J. Kiefer,
and K. Krickeberg
2
Mathematical Statistics
and Probability Theory
Proceedings, Sixth International Conference,
WisYa (Poland), 1978
Edited by W. Klonecki, A. Kozek,
and J. Rosinski
Springer-Verlag
New York Heidelberg Berlin
Editors
Dr. Witold Klonecki, Dr. Andrzej Kozek. Dr. Jan Rosinski
Mathematical Institute of the Polish Academy of Science
Kopernika 18. 51-617 Wroclaw. Poland
AMS Subject Classifications: 62-XX
Library of Congress Cataloging in Publication Data
Main entry under title:
Mathematical statistics and probability theory.
(Lecture notes in statistics; 2)
Includes bibliographies
1. Mathematical statistics-Congresses.
2. Probabilities-Congresses. I. Klonecki. Witold.
II. Kozek. A. III. Rosinski. Jan. IV. Series.
QA276.A1M3 519.5 80-13322
ISBN 978-0-387-90493-1 ISBN 978-1-4615-7397-5 (eBook)
DOI 10.1007/978-1-4615-7397-5
All rights reserved.
No part of this book may be translated or reproduced in any form without
written permission from Springer-Verlag.
© 1980 by Springer-Verlag New York Inc.
987654321
Oed icated to
Professor Jerzy Neyman
FOREWORD
Since 1972 the Institute of Mathematics and the Committee of
Mathematics of the Polish Academy of Sciences organize annually con
ferences on mathematical statistics in Wisla. The 1978 conference,
supported also by the University of Wroclaw,was held in Wisla from
December 7 to December 13 and attended by around 100 participants
from 11 countries. K. Urbanik, Rector of the University of Wroclaw,
was the honorary chairman of the conference.
Traditionally at these conferences there are presented results on
mathematical statistics and related fields obtained in Poland during
the year of the conference as well as results presented by invited
scholars from other countries. In 1978 invitations to present talks
were accepted by 20 e~inent statisticians and probabilists. The
topics of the invited lectures and contributed papers included
theoretical statistics with a broad cover of the theory of linear
models, inferences from stochastic processes, probability theory
and applications to biology and medicine. In these notes there
appear papers submitted by 30 participants of the conference.
During the conference, on December 9, there was held a special
session of the Polish Mathematical Society on the occasion of elect
ing Professor Jerzy Neyman the honorary member of the Polish
Mathematical Society. At this session W. Orlicz, president of the
Polish Mathematical Society, K.Krickeberg,president of the Bernoulli
Society. R. Bartoszynski and K. Doksum gave talks on Neyman IS con
tribution to statistics, his organizational achievements in the U.S.
and his role as a founder of the IASPS, the forerunner of the Ber
noulli Society~hreeof the talks appear in this volume).
V
VI
We would like to thank all lecturers, including those at the
session of the Polish Mathematical Society, all chairmen and
participants for the contributions.
The organization of the conference was in very capable hands
of Mrs. A. Huskowski and Mr. E. Mordzinski.
w.
Klonecki
A. Kozak
J. Rosinski
CONTENTS
R. BartoszyD.ski
SOME THOUGHTS ABOUT JERZY NEYMAN XI
K. Doksum
SOME REMARKS ON THE ACHIEVEMENTS OF
PROFESSOR NEYMAN IN THE UNITED STATES XVII
K. Krickeberg
ROLE OF JERZY NEYMAN IN THE SHAPING OF THE
BERNOULLI SOCIETY xx
O. AALEN
A Model for Nonparametric Regression Analysis of
Counting Processes 1
R. BANYS
On Superpositions of Random Measures and Point
Processes 26
T. BEDNARSKI
Application and Optimality of the Chi-Square Test of
&-
Fit for Testing Validity of Parametric Models 38
,
T. CALINSKI, B. CERANKA, S. MEJZA
On the Notion of Efficiency of a Block Design 47
D. M. CHIBISOV
An Asymptotic Expansion for Distributions of
C (a) Test Statistics 63
Z. CIESIELSKI
Properties of Realizations of Random Fields 97
, ,
J. CWIK, T. KOWALCZYK, A. KOWALSKI, E. PLESZCZYNSKA,
W. SZCZESNY, T. WIERZBOWSKA
Monotone Dependence Function: Background, New
Results and Applications 111
K. A. DOKSUM
Lifetesting for Matched Pairs 122
N. GAFFKE, O. KRAFFT
D-Optimum Designs for the Interblock-Model 134
S. GNOT
Locally Best Linear Estimation in Euclidean
Vector Spaces 144
B. GRIGELIONIS, R. MIKULEVICIUS
On Statistical Problems of Stochastic Processes with
Penetrable Boundaries 152
VII
VIII
P. HELLMANN
On TWo-Sided Nonparametric Tests for the TWo-Sample
Problem 170
A. JAKUBOWSKI
On Limit Theorems for Sums of Dependent Hilbert
Space Valued Random Variables 178
J. KLEFFE
C. R. Rao's MINQUE for Replicated and Multivariate
Observations 188
W. KLONECKI
Invariant Quadratic Unbiased Estimation for
Variance Components 201
A. KLOPOTOWSKI
Mixtures of Infinitely Divisible Distributions as
Limit Laws for Sums of Dependent Random Variables 224
A. KOZEK, Z. SUCHANECKI
Conditional Expectations of Selectors and Jensen's
Inequality 247
L. R. LAMOTTE
Some Results on Biased Linear Estimation Applied to
Variance Component Estimation 266
R. MAGIERA
Estimation Problem for the Exponential Class
of Distributions from Delayed Observations 275
D. MAJUMDAR, S. K. MITRA
Statistical Analysis of Nonestimable Functionals.
Part I: Estimation 288
D. MAJUMDAR, S. K. MITRA
A Correcting Note to "Statistical Analysis of
Nonestimable Functions. Part I: Estimation" 317
,
M. MUS lELA , R. ZMYSLONY
Estimation for Some Classes of Gaussian Markov Processes 318
M. MUSIELA, R. ZMYSLONY
Estimation of Regression Parameters of Gaussian Markov
Processes 330
,
J. ROSINSKI
Some Remarks on the Central Limit Theorem in Branch
Spaces 342
V. I. TARIELADZE
Characterization of Covariance Operators Which
Guarantee the CLT 348
,
R. ZIELINSKI
Fixed Precision Estimate of Mean of a Gaussian
Sequence with Unknown Covariance Structure 360
,
R. ZMYSLONY
A Characterization of Best Linear Unbiased
Estimators in the General Linear Model 365
SOME THOUGHTS ABOUT JERZY NEYMAN
by
Robert Bartoszynski
It is the seoond time within the last few years that I have the
honour and privilege to have a talk about Professor Neyman and his
contribution to statistios.
Let me start with few words of explanation of Neyman's biography.
We heard from the speech of Professor Orlicz that Neyman is
a grandson of an insurgent of 1863. Now, this information carries
a very clear meaning to all Poles, but may perhaps be somewhat
puzzling to non-Poles.
The point is that the Uprising of 1863 is a sort of holy eve.nt
in the Polish history, and the knowledge that someone comes from
a family whose members took part in it makes him automatioally
somehow dearer to the Poles.
Incidentally, I had been discussing with my oolleagues whether
or not such information about Neyman ought to be included in his
biography. Not that anyone wanted to make Neyman less dear to us, of
course; the question was: does Neyman really need that kind of
"support"? His greatness comes from what he himself has done,and not
from the merits of his family. Anyway, I am quite happy that the
problem was resolved for me by Professor Orlicz, and that I oould
give these few words of explanation.
Now, Neyman's oontributions to statistics are well known,and not
likely to be underestimated by anyone who has any understanding of
statistios. To put it most briefly, they consist of stating for the
IX
x
first time (together with E.S. Pearson) the principles of testing
hypotheses, with the crucial concept of the power of the test;
introducing the notion of confidence interval; and formulating the
principles of optimization in sampling theory.
All this,as I said, is well known, and I repeat it merely because
without mentioning these facts any talk about Neyman's contribution
would not be complete.
What I wanted to present in some more detail today, are just two
examples of statements of some problems connected closely with
empirical domains.The aim is simply to illustrate the art - in which
Neyman excels - of transforming the real-life problems into statisti-
cal ones.
The first of these problems conoerns the so-called outliers [3].
An outlier is, roughly speaking, an element in the sample whioh is
larger (say) than the remaining elements, to such a degree that one
wonders if it is a genuine sample element, or perhaps results from
an error of observation or error in recording the data.
To put it formally, let Y1' Y2' ••• ' Yn be a sample from some
underlying distribution F. We assume therefore that Yi's are
independent and F(t) = P(Yi ~ t), i = 1, ••• ,n. Assume further that
F has a density f; we may then neglect the possibility of ties among
elements of the sample. Let x1<x2< ••• <xn be the sample Y1'Y2' ••• Yn
ordered aooording to magnitude.
Suppose that xn is an element which" appears too large"; we
would like to devise a test for the hypothesis that it comes from
the same distribution as the rest of the sample. It is intuitively
clear that any test of such a hypothesis should be based on compari
son of the "distance" from xn to the "group" x1, ••• ,xn-1' with the
"spread" of this group. The terms "distance" and "spread" were put
in the quotation marks to stress the fact that the test statistios
may be oonstructed in a number of ways, by taking as a distanoe