Lectures in Mathematical Statistics, Parts 1 and 2 (AMS Translations of Mathematical Monographs, Volume 229)

333 Pages·2005·12.31 MB·English

by  Yu. N. Linkov

Translations of
m a t h e m a t i c a l
M O N O G R APH S
Volume 229
Lectures in
Mathematical
Statistics
Parts 1 and 2
Yu. N. Lin'kov
American Mathematical Society
Lectures in
Mathematical
Statistics
Parts 1 and 2

EDITORIAL COMMITTEE
AMS Subcommittee
Robert D. MacPherson Grigorii A. Margulis James D. StashefF (Chair)
ASL Subcommittee Steffen Lempp (Chair)
IMS Subcommittee Mark I. Preidlin (Chair)
Ю. H. Линьков
ЛЕКЦИИ ПО МАТЕМАТИЧЕСКОЙ СТАТИСТИКЕ
“ИСТОКИ”, ДОНЕЦК, 2001
This work was originally published in Russian by Istoki, Donetsk under the title
“Лекции no математической статистике, части 1,2” © Ю. Н. Линьков, 1999. The
present translation was created under license for the American Mathematical Society and
is published by permission.
Ti’anslated from the Russian by Oleg Klesov and Vladimir Zayats.
2000 Mathematics Subject Classification. Primary 62-01.
For additional information and updates on this book, visit
WWW. ams. org / b ookpages / mmono- 229
Library of Congress Cataloging-in-Publication Data
Lin'kov, lU. N.
[Lektsii po matematicheskoi statistike. English]
Lectures in mathematical statistics : parts 1 and 2 / Yu. N. Lin'kov ; translated by Oleg Klesov
and Vladimir Zayats.
p. cm. - (Translations of mathematical monographs, ISSN 0065-9282 ; v. 229)
Includes bibliographical references and index.
ISBN 0-8218-3732-X (alk. paper)
1. Mathematical statistics. I. Title II. Series
QA276.16.L5513 2005
519.5-dc22 2005052661
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
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Republication, systematic copying, or multiple reproduction of any material in this publication
is permitted only under license from the American Mathematical Society. Requests for such
permission should be addressed to the Acquisitions Department, American Mathematical Society,
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© 2005 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
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established to ensure permanence and durability.
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10 9 8 7 6 5 4 3 2 1 10 09 08 07 06 05
Contents
Foreword to the English Translation vii
Part 1 1
Preface to Part 1 3
Chapter 1. Samples from One-Dimensional Distributions 5
1.1. Empirical distribution function and its asymptotic behavior 5
1.2. Sample characteristics and their properties 8
1.3. Order statistics and their properties 13
1.4. The distributions of some functions of Gaussian random vectors 20
Chapter 2. Samples from Multidimensional Distributions 25
2.1. Empirical distribution function, sampling moments, and their
properties 25
2.2. Sampling regression and its properties 31
Chapter 3. Estimation of Unknown Parameters of Distributions 39
3.1. Statistical estimators and their quality measures 39
3.2. Estimation of a location parameter 49
3.3. Estimation of a scale parameter 56
3.4. The Cramer-Rao inequality and efficient estimators 61
3.5. The Cramer-Rao inequality for a multidimensional parameter 80
3.6. Integral inequalities of Cramer-Rao type 88
Chapter 4. Sufficient Statistics 99
4.1. Sufficient statistics and a theorem on factorization 99
4.2. Sufficient statistics and optimal estimators 113
Chapter 5. General Methods for Constructing Estimators 131
5.1. Method of moments 131
5.2. The maximum likelihood method 133
5.3. Bayes and minimax methods 142
5.4. Confidence intervals and regions 147
References to Part 1 153
Vi CONTENTS
Part 2 155
Preface to Part 2 157
Chapter 1. General Theory of Hypotheses Testing 159
1.1. Testing two simple hypotheses 159
1.2. Distinguishing a finite number of simple hypotheses 173
1.3. Distinguishing composite hypotheses 182
Chapter 2. Asymptotic Distinguishability of Simple Hypotheses 203
2.1. Statistical hypotheses and tests 203
2.2. Types of the asymptotic distinguishability of families of hypothe­
ses. The characterization of types 205
2.3. Complete asymptotic distinguishability under the strong law of
large numbers 218
2.4. Complete asymptotic distinguishability under the weak conver­
gence 238
2.5. Contiguous families of hypotheses 248
Chapter 3. Goodness-of-Fit Tests 263
3.1. The setting of the problem. Kolmogorov test 263
3.2. The Pearson test 266
3.3. Smirnov test 275
3.4. Other goodness-of-fit tests 282
Chapter 4. Sequential Tests 293
4.1. Bayes sequential tests of hypotheses 293
4.2. Wald sequential tests 300
4.3. The optimality of a sequential Wald test 310
References to Part 2 317
Index 319
Foreword to the English Translation
Parts 1 and 2 of “Lectures in Mathematical Statistics” by Yu. N. Lin'kov were
originally published in Russian as two separate books. For the English translation,
the two parts are combined into one book. Each part has its own preface and list
of references, with chapters, sections, theorems, etc., numbered independently in
each part.

Part 1