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Springer Texts in Statistics
Advisors:
George Casella Stephen Fienberg Ingram Olkin
E.L. Lehmann Joseph P. Romano
Testing Statistical
Hypotheses
Third Edition
With 6 Illustrations
E.L. Lehmann
Professor of Statistics Emeritus
Department of Statistics
University of California, Berkeley
Berkeley, CA 94720
USA
Joseph P. Romano
Department of Statistics
Stanford University
Sequoia Hall
Stanford, CA 94305
USA
romano@stanford.edu
Editorial Board
George Casella
Stephen Fienberg
Ingram Olkin
Department of Statistics
University of Florida
Gainesville, FL 32611-8545
USA
Department of Statistics
Carnegie Mellon University
Pittsburgh, PA 15213-3890
USA
Department of Statistics
Stanford University
Stanford, CA 94305
USA
Library of Congress Cataloging-in-Publication Data
A catalog record for this book is available from the Library of Congress.
ISBN 0-387-98864-5
Printed on acid-free paper.
© 2005, 1986, 1959 Springer Science+Business Media, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New
York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis.
Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if
they are not identified as such, is not to be taken as an expression of opinion as to whether or not
they are subject to proprietary rights.
Printed in the United States of America.
9 8 7 6 5 4 3 2 1
SPIN 10728642
Typesetting: Pages created by the author.
springeronline.com
(MVY)
Dedicated to the Memory of
Lucien Le Cam (1924-2000) and John W. Tukey (1915-2000)
Preface to the Third Edition
The Third Edition of Testing Statistical Hypotheses brings it into consonance
with the Second Edition of its companion volume on point estimation (Lehmann
and Casella, 1998) to which we shall refer as TPE2. We won’t here comment on
the long history of the book which is recounted in Lehmann (1997) but shall use
this Preface to indicate the principal changes from the 2nd Edition.
The present volume is divided into two parts. Part I (Chapters 1–10) treats
small-sample theory, while Part II (Chapters 11–15) treats large-sample theory.
The preface to the 2nd Edition stated that “the most important omission is an
adequate treatment of optimality paralleling that given for estimation in TPE.”
We shall here remedy this failure by treating the di?cult topic of asymptotic
optimality (in Chapter 13) together with the large-sample tools needed for this
purpose (in Chapters 11 and 12). Having developed these tools, we use them in
Chapter 14 to give a much fuller treatment of tests of goodness of ?t than was
possible in the 2nd Edition, and in Chapter 15 to provide an introduction to
the bootstrap and related techniques. Various large-sample considerations that
in the Second Edition were discussed in earlier chapters now have been moved to
Chapter 11.
Another major addition is a more comprehensive treatment of multiple testing
including some recent optimality results. This topic is now presented in Chapter
9. In order to make room for these extensive additions, we had to eliminate some
material found in the Second Edition, primarily the coverage of the multivariate
linear hypothesis.
Except for some of the basic results from Part I, a detailed knowledge of smallsample theory is not required for Part II. In particular, the necessary background
should include: Chapter 3, Sections 3.1–3.5, 3.8–3.9; Chapter 4: Sections 4.1–4.4;
Chapter 5, Sections 5.1–5.3; Chapter 6, Sections 6.1–6.2; Chapter 7, Sections
7.1–7.2; Chapter 8, Sections 8.1–8.2, 8.4–8.5.
viii
Preface
Of the two principal additions to the Third Edition, multiple comparisons
and asymptotic optimality, each has a godfather. The development of multiple
comparisons owes much to the 1953 volume on the subject by John Tukey, a
mimeographed version which was widely distributed at the time. It was o?cially
published only in 1994 as Volume VIII in The Collected Works of John W. Tukey.
Many of the basic ideas on asymptotic optimality are due to the work of Le
Cam between 1955 and 1980. It culminated in his 1986 book, Asymptotic Methods
in Statistical Decision Theory.
The work of these two authors, both of whom died in 2000, spans the achievements of statistics in the second half of the 20th century, from model-free
data analysis to the most abstract and mathematical asymptotic theory. In acknowledgment of their great accomplishments, this volume is dedicated to their
memory.
Special thanks to Noureddine El Karoui, Matt Finkelman, Brit Katzen, Mee
Young Park, Elizabeth Purdom, Armin Schwartzman, Azeem Shaikh and the
many students at Stanford University who proofread several versions of the new
chapters and worked through many of the over 300 new problems. The support
and suggestions of our colleagues is greatly appreciated, especially Persi Diaconis, Brad Efron, Susan Holmes, Balasubramanian Narasimhan, Dimitris Politis,
Julie Sha?er, Guenther Walther and Michael Wolf. Finally, heartfelt thanks go to
friends and family who provided continual encouragement, especially Ann Marie
and Mark Hodges, David Fogle, Scott Madover, David Olachea, Janis and Jon
Squire, Lucy, and Ron Susek.
E. L. Lehmann
Joseph P. Romano
January, 2005
Contents
Preface
I
vii
Small-Sample Theory
1
1 The
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
General Decision Problem
Statistical Inference and Statistical Decisions
Speci?cation of a Decision Problem . . . . .
Randomization; Choice of Experiment . . .
Optimum Procedures . . . . . . . . . . . . .
Invariance and Unbiasedness . . . . . . . . .
Bayes and Minimax Procedures . . . . . . .
Maximum Likelihood . . . . . . . . . . . . .
Complete Classes . . . . . . . . . . . . . . .
Su?cient Statistics . . . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . . .
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3
3
4
8
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11
14
16
17
18
21
27
2 The
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Probability Background
Probability and Measure . . . . . . . . .
Integration . . . . . . . . . . . . . . . . .
Statistics and Sub?elds . . . . . . . . . .
Conditional Expectation and Probability
Conditional Probability Distributions . .
Characterization of Su?ciency . . . . . .
Exponential Families . . . . . . . . . . .
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31
34
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41
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46
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x
Contents
2.8
2.9
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Uniformly Most Powerful Tests
3.1
Stating The Problem . . . . . . . . . . . . . .
3.2
The Neyman–Pearson Fundamental Lemma .
3.3
p-values . . . . . . . . . . . . . . . . . . . . .
3.4
Distributions with Monotone Likelihood Ratio
3.5
Con?dence Bounds . . . . . . . . . . . . . . .
3.6
A Generalization of the Fundamental Lemma
3.7
Two-Sided Hypotheses . . . . . . . . . . . . .
3.8
Least Favorable Distributions . . . . . . . . .
3.9
Applications to Normal Distributions . . . . .
3.9.1 Univariate Normal Models . . . . . . .
3.9.2 Multivariate Normal Models . . . . . .
3.10 Problems . . . . . . . . . . . . . . . . . . . . .
3.11 Notes . . . . . . . . . . . . . . . . . . . . . . .
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4 Unbiasedness: Theory and First Applications
4.1
Unbiasedness For Hypothesis Testing . . . . . . . . .
4.2
One-Parameter Exponential Families . . . . . . . . .
4.3
Similarity and Completeness . . . . . . . . . . . . . .
4.4
UMP Unbiased Tests for Multiparameter Exponential
4.5
Comparing Two Poisson or Binomial Populations . .
4.6
Testing for Independence in a 2 × 2 Table . . . . . .
4.7
Alternative Models for 2 × 2 Tables . . . . . . . . . .
4.8
Some Three-Factor Contingency Tables . . . . . . . .
4.9
The Sign Test . . . . . . . . . . . . . . . . . . . . . .
4.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . .
4.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
55
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56
56
59
63
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72
77
81
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86
89
92
107
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Families
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110
110
111
115
119
124
127
130
132
135
139
149
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5 Unbiasedness: Applications to Normal Distributions
150
5.1
Statistics Independent of a Su?cient Statistic . . . . . . . . .
150
5.2
Testing the Parameters of a Normal Distribution . . . . . . .
153
5.3
Comparing the Means and Variances of Two Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
157
5.4
Con?dence Intervals and Families of Tests . . . . . . . . . . .
161
5.5
Unbiased Con?dence Sets . . . . . . . . . . . . . . . . . . . . .
164
5.6
Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
168
5.7
Bayesian Con?dence Sets . . . . . . . . . . . . . . . . . . . . .
171
5.8
Permutation Tests . . . . . . . . . . . . . . . . . . . . . . . . .
176
5.9
Most Powerful Permutation Tests . . . . . . . . . . . . . . . .
177
5.10 Randomization As A Basis For Inference . . . . . . . . . . . .
181
5.11 Permutation Tests and Randomization . . . . . . . . . . . . .
184
5.12 Randomization Model and Con?dence Intervals . . . . . . . .
187
5.13 Testing for Independence in a Bivariate Normal Distribution .
190
5.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
192
5.15 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
210
Contents
6 Invariance
6.1
Symmetry and Invariance . . . . . . . . . . . .
6.2
Maximal Invariants . . . . . . . . . . . . . . .
6.3
Most Powerful Invariant Tests . . . . . . . . .
6.4
Sample Inspection by Variables . . . . . . . .
6.5
Almost Invariance . . . . . . . . . . . . . . . .
6.6
Unbiasedness and Invariance . . . . . . . . . .
6.7
Admissibility . . . . . . . . . . . . . . . . . . .
6.8
Rank Tests . . . . . . . . . . . . . . . . . . . .
6.9
The Two-Sample Problem . . . . . . . . . . .
6.10 The Hypothesis of Symmetry . . . . . . . . .
6.11 Equivariant Con?dence Sets . . . . . . . . . .
6.12 Average Smallest Equivariant Con?dence Sets
6.13 Con?dence Bands for a Distribution Function
6.14 Problems . . . . . . . . . . . . . . . . . . . . .
6.15 Notes . . . . . . . . . . . . . . . . . . . . . . .
xi
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212
212
214
218
223
225
229
232
239
242
246
248
251
255
257
276
7 Linear Hypotheses
7.1
A Canonical Form . . . . . . . . . . . . . . . . .
7.2
Linear Hypotheses and Least Squares . . . . . .
7.3
Tests of Homogeneity . . . . . . . . . . . . . . .
7.4
Two-Way Layout: One Observation per Cell . .
7.5
Two-Way Layout: m Observations Per Cell . . .
7.6
Regression . . . . . . . . . . . . . . . . . . . . .
7.7
Random-E?ects Model: One-way Classi?cation .
7.8
Nested Classi?cations . . . . . . . . . . . . . . .
7.9
Multivariate Extensions . . . . . . . . . . . . . .
7.10 Problems . . . . . . . . . . . . . . . . . . . . . .
7.11 Notes . . . . . . . . . . . . . . . . . . . . . . . .
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277
277
281
285
287
290
293
297
300
304
306
317
8 The
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
Minimax Principle
Tests with Guaranteed Power . . . . . . .
Examples . . . . . . . . . . . . . . . . . . .
Comparing Two Approximate Hypotheses
Maximin Tests and Invariance . . . . . . .
The Hunt–Stein Theorem . . . . . . . . . .
Most Stringent Tests . . . . . . . . . . . .
Problems . . . . . . . . . . . . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . .
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319
319
322
326
329
331
337
338
347
9 Multiple Testing and Simultaneous Inference
9.1
Introduction and the FWER . . . . . . . . . .
9.2
Maximin Procedures . . . . . . . . . . . . . .
9.3
The Hypothesis of Homogeneity . . . . . . . .
9.4
Sche?e´’s S-Method: A Special Case . . . . . .
9.5
Sche?e´’s S-Method for General Linear Models
9.6
Problems . . . . . . . . . . . . . . . . . . . . .
9.7
Notes . . . . . . . . . . . . . . . . . . . . . . .
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348
348
354
363
375
380
385
391
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xii
Contents
10 Conditional Inference
10.1 Mixtures of Experiments .
10.2 Ancillary Statistics . . . .
10.3 Optimal Conditional Tests
10.4 Relevant Subsets . . . . .
10.5 Problems . . . . . . . . . .
10.6 Notes . . . . . . . . . . . .
II
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Large-Sample Theory
392
392
395
400
404
409
414
417
11 Basic Large Sample Theory
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Basic Convergence Concepts . . . . . . . . . . . . . . . .
11.2.1 Weak Convergence and Central Limit Theorems
11.2.2 Convergence in Probability and Applications . . .
11.2.3 Almost Sure Convergence . . . . . . . . . . . . .
11.3 Robustness of Some Classical Tests . . . . . . . . . . . .
11.3.1 E?ect of Distribution . . . . . . . . . . . . . . . .
11.3.2 E?ect of Dependence . . . . . . . . . . . . . . . .
11.3.3 Robustness in Linear Models . . . . . . . . . . . .
11.4 Nonparametric Mean . . . . . . . . . . . . . . . . . . . .
11.4.1 Edgeworth Expansions . . . . . . . . . . . . . . .
11.4.2 The t-test . . . . . . . . . . . . . . . . . . . . . .
11.4.3 A Result of Bahadur and Savage . . . . . . . . .
11.4.4 Alternative Tests . . . . . . . . . . . . . . . . . .
11.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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419
419
424
424
431
440
444
444
448
451
459
459
462
466
468
469
480
12 Quadratic Mean Di?erentiable Families
12.1 Introduction . . . . . . . . . . . . . . . . .
12.2 Quadratic Mean Di?erentiability (q.m.d.) .
12.3 Contiguity . . . . . . . . . . . . . . . . . .
12.4 Likelihood Methods in Parametric Models
12.4.1 E?cient Likelihood Estimation . .
12.4.2 Wald Tests and Con?dence Regions
12.4.3 Rao Score Tests . . . . . . . . . . .
12.4.4 Likelihood Ratio Tests . . . . . . .
12.5 Problems . . . . . . . . . . . . . . . . . . .
12.6 Notes . . . . . . . . . . . . . . . . . . . . .
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13 Large Sample Optimality
13.1 Testing Sequences, Metrics, and Inequalities
13.2 Asymptotic Relative E?ciency . . . . . . . .
13.3 AUMP Tests in Univariate Models . . . . .
13.4 Asymptotically Normal Experiments . . . .
13.5 Applications to Parametric Models . . . . .
13.5.1 One-sided Hypotheses . . . . . . . .
13.5.2 Equivalence Hypotheses . . . . . . .
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14 Testing Goodness of Fit
14.1 Introduction . . . . . . . . . . . . . . . . . . .
14.2 The Kolmogorov-Smirnov Test . . . . . . . . .
14.2.1 Simple Null Hypothesis . . . . . . . . .
14.2.2 Extensions of the Kolmogorov-Smirnov
14.3 Pearson’s Chi-squared Statistic . . . . . . . .
14.3.1 Simple Null Hypothesis . . . . . . . . .
14.3.2 Chi-squared Test of Uniformity . . . .
14.3.3 Composite Null Hypothesis . . . . . .
14.4 Neyman’s Smooth Tests . . . . . . . . . . . .
14.4.1 Fixed k Asymptotics . . . . . . . . . .
14.4.2 Neyman’s Smooth Tests With Large k
14.5 Weighted Quadratic Test Statistics . . . . . .
14.6 Global Behavior of Power Functions . . . . . .
14.7 Problems . . . . . . . . . . . . . . . . . . . . .
14.8 Notes . . . . . . . . . . . . . . . . . . . . . . .
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15 General Large Sample Methods
15.1 Introduction . . . . . . . . . . . . . . .
15.2 Permutation and Randomization Tests
15.2.1 The Basic Construction . . . . .
15.2.2 Asymptotic Results . . . . . . .
15.3 Basic Large Sample Approximations .
15.3.1 Pivotal Method . . . . . . . . .
15.3.2 Asymptotic Pivotal Method . .
15.3.3 Asymptotic Approximation . .
15.4 Bootstrap Sampling Distributions . . .
15.4.1 Introduction and Consistency .
15.4.2 The Nonparametric Mean . . .
15.4.3 Further Examples . . . . . . . .
15.4.4 Stepdown Multiple Testing . . .
15.5 Higher Order Asymptotic Comparisons
15.6 Hypothesis Testing . . . . . . . . . . .
15.7 Subsampling . . . . . . . . . . . . . . .
15.7.1 The Basic Theorem in the I.I.D.
15.7.2 Comparison with the Bootstrap
15.7.3 Hypothesis Testing . . . . . . .
15.8 Problems . . . . . . . . . . . . . . . . .
15.9 Notes . . . . . . . . . . . . . . . . . . .
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A Auxiliary Results
A.1 Equivalence Relations; Groups . . . . . . . . . . . . . . . . . .
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13.5.3 Multi-sided Hypotheses . . . . . . . . .
Applications to Nonparametric Models . . . .
13.6.1 Nonparametric Mean . . . . . . . . . .
13.6.2 Nonparametric Testing of Functionals .
Problems . . . . . . . . . . . . . . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . . . .
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xiv
Contents
A.2
A.3
A.4
A.5
Convergence of Functions; Metric Spaces
Banach and Hilbert Spaces . . . . . . . .
Dominated Fa …
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