Table Of ContentNonparametric Estimation under Shape
Constraints
This book treats the latest developments in the theory of order-restricted inference, with
special attention to nonparametric methods and algorithmic aspects. Among the topics
treated are current status and interval censoring models, competing risk models and
deconvolution. Methods of order-restricted inference are used in computing maximum
likelihood estimators and developing distribution theory for inverse problems of this type.
The authors have been active in developing these tools and present the state of the art
and open problems in the field. The earlier chapters provide an introduction to the subject,
while the later chapters are written with graduate students and researchers in mathematical
statistics in mind. Each chapter ends with a set of exercises of varying difficulty. The
theory is illustrated with the analysis of real-life data, which are mostly medical in nature.
PIET GROENEBOOM is Professor Emeritus of Statistics at Delft University of Technology,
the Netherlands.
GEURT JONGBLOED is Professor of Statistics at Delft University of Technology, the
Netherlands.
CAMBRIDGE SERIES IN STATISTICAL AND
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M. Stein (Department of Statistics, University of Chicago)
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For a list of books in the series, see end of book.
Nonparametric Estimation under Shape
Constraints
Estimators, Algorithms and Asymptotics
Piet Groeneboom
Delft University of Technology
Geurt Jongbloed
Delft University of Technology
32 Avenue of the Americas, New York, NY 10013-2473, USA
Cambridge University Press is part of the University of Cambridge.
It furthers the University’s mission by disseminating knowledge in the pursuit of
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Information on this title: www.cambridge.org/9780521864015
© Piet Groeneboom and Geurt Jongbloed 2014
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2014
Printed in the United States of America
A catalog record for this publication is available from the British Library.
Library of Congress Cataloging in Publication Data
Groeneboom, P.
Nonparametric estimation under shape constraints : estimators, algorithms and
asymptotics / Piet Groeneboom, Technische Universiteit Delft, The Netherlands, Geurt
Jongbloed, Technische Universiteit Delft, The Netherlands. pages cm. – (Cambridge
series in statistical and probabilistic mathematics ; 38) Includes bibliographical references
and index.
ISBN 978-0-521-86401-5 (hbk.)
1. Nonparametric statistics. 2. Multivariate analysis. 3. Mathematical statistics. 4.
Estimation theory.
I. Jongbloed, Geurt, 1968– II. Title.
QA278.8.G76 2014 519.5′4–dc23 2014022148
ISBN 978-0-521-86401-5 Hardback
Additional resources for this publication at http://statistics.tudelft.nl/CUPbook
Cambridge University Press has no responsibility for the persistence or accuracy of URLs
for external or third-party Internet websites referred to in this publication and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents
Preface and Acknowledgments
1 Introduction
1.1 Is There a Warming-up of Lake Mendota?
1.2 Onset of Nonlethal Lung Tumor
1.3 The Transmission Potential of a Disease
1.4 The Bangkok Cohort Study
1.5 Inverse Problems, Censoring, Mixture Models and Shape Constraints
1.6 Outline of the Book
Exercises
Bibliographic Remarks
2 Basic Estimation Problems with Monotonicity Constraints
2.1 Monotone Regression
2.2 Monotone Density Estimation
2.3 Estimating a Distribution Function from Current Status Data
2.4 Deconvolution Problem with Jump Kernel
2.5 Generalized Isotonic Regression Problems
2.6 Estimating a Monotone Hazard Rate
Exercises
Bibliographic Remarks
3 Asymptotic Theory for the Basic Monotone Problems
3.1 Consistency
3.2 Heuristic Asymptotics for the Grenander Estimator
3.3 Convex Minorants: Basic Properties
3.4 Some Empirical Process Theory
3.5 Asymptotic Distribution in Exponential Deconvolution Model
3.6 Limit Distribution of the Grenander Estimator
3.7 Some Martingale Theory
3.8 Asymptotic Distribution of the MLE in the Current Status Model
3.9 Chernoff’s Distribution
3.10 The Concave Majorant of Brownian Motion and Brownian Bridge
Exercises
Bibliographic Remarks
4 Other Univariate Problems Involving Monotonicity Constraints
4.1 Wicksell’s Corpuscle Problem
4.2 Convex Regression
4.3 Convex Density Estimation
4.4 Log Concave Densities
4.5 Star Shaped Distributions on [0, 1]
4.6 Deconvolution Problems
4.7 Interval Censoring Case 2
Exercises
Bibliographic Remarks
5 Higher Dimensional Problems
5.1 Competing Risks with Current Status Observations
5.2 Bivariate Interval Censoring
5.3 Current Status with Continuous Marks
5.4 Multivariate Log Concave Densities
Exercises
Bibliographic Remarks
6 Lower Bounds on Estimation Rates
6.1 Global and Local Minimax Risk
6.2 A Minimax Lower Bound Theorem
6.3 Lower Bound Based on the Van Trees Inequality
6.4 Applications
Exercises
Bibliographic Remarks
7 Algorithms and Computation
7.1 Algorithm: Concept and Convergence
7.2 The EM Algorithm
7.3 The Iterative Convex Minorant Algorithm
7.4 Vertex Direction Algorithms
7.5 The MLE in the Competing Risks Model with Interval Censoring
Exercises
Bibliographic Remarks
8 Shape and Smoothness
8.1 Smoothing a Shape-Constrained Estimator
8.2 Maximizing a Smoothed Objective Function
8.3 Penalized M-Estimation
8.4 Monotonic Rearrangements of Smooth Estimators
8.5 Maximum Smoothed Likelihood Estimation in the Current Status Model
8.6 Smooth Estimation for Interval Censoring Case 2
8.7 Smooth Estimation in the Bivariate Interval Censoring Model
Exercises
Bibliographic Remarks
9 Testing and Confidence Intervals
9.1 Testing for a Monotone Hazard
9.2 k-Sample Tests for Decreasing Densities
9.3 Two-Sample Tests for Current Status Data
9.4 Two-Sample Tests for Case 2 Interval Censored Data
9.5 Pointwise Confidence Intervals for the Current Status Model
Exercises
Bibliographic Remarks
10 Asymptotic Theory of Smooth Functionals
10.1 Estimating the Expectation in Deconvolution Models
10.2 Estimating Smooth Functionals in the Current Status Model
10.3 The Integral Equation for Interval Censoring Case 2
10.4 Smooth Functional Estimation in the Interval Censoring Case 2 Model
Exercises
Bibliographic Remarks
11 Pointwise Asymptotic Distribution Theory for Univariate Problems
11.1 The LS Estimator of a Convex Density
11.2 Tail Bounds for the MLE in the Current Status Model
11.3 The SMLE in the Current Status Model
11.4 The SMLE for Interval Censoring Case 2
11.5 The MSLE for Interval Censoring Case 2
11.6 Estimation of a Nondecreasing Hazard in the Right Censoring Model: SMLE
Exercises
Bibliographic Remarks
12 Pointwise Asymptotic Distribution Theory for Multivariate Problems
12.1 The ML Estimator in the Competing Risk Model with Current Status Data
12.2 The SMLE in the Current Status Competing Risk Model
12.3 The Bivariate Current Status Model
Exercises
Bibliographic Remarks
13 Asymptotic Distribution of Global Deviations
13.1 The L Loss of the Grenander Estimator
1
13.2 Empirical L -Test for a Monotone Hazard
1
13.3 Two-Sample Tests for Current Status Data
Exercises
Bibliographic Remarks
References
Author Index
Subject Index
