Nonparametric Goodness-of-Fit Testing under Gaussian Models by Yu I. Ingster; Irina A.SuslinaReview by: Christophe PouetJournal of the American Statistical Association, Vol. 99, No. 466 (Apr., 2004), pp. 561-562Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/27590415 .

Accessed: 14/06/2014 13:27

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journalof the American Statistical Association.

http://www.jstor.org

This content downloaded from 195.34.79.253 on Sat, 14 Jun 2014 13:27:00 PMAll use subject to JSTOR Terms and Conditions

Book Reviews

recent developments that move away from hypothesis tests toward model selec

tion and maximum likelihood, Bayesian, and approximate Bayesian inference

of parameters are absent. Although these topics are fairly new to population

genetics, empirical investigators who wish to answer the question posed in the

Preface?"given a collection of DNA sequences, what underlying forces are

responsible for the observed patterns of variability?"?would be interested to

know about them. Finally, although examples are used effectively to describe

individual methods, some synthesis that connects several techniques to the same

example dataset would help empirical population geneticists design analyses of

their own data.

Despite these limitations, if used in conjunction with texts that state results

in broader biological context, this book would be helpful to those with a math

ematical or statistical background who are encountering the subject for the first

time. The book is more theoretical than most books in the area, and because it is

more modern than related classics, such as those by Crow and Kimura (1970) and Ewens (1979), it will be an informative reference for researchers in the

field.

Noah A. Rosenberg

University of Southern California

REFERENCES

Crow, J. F., and Kimura, M. (1970), An Introduction to Population Genetics

Theory, Minneapolis: Burgess. Ewens, W. J. (1979), Mathematical Population Genetics. Berlin: Springer

Verlag. Hartl, D. L., and Clark, A. G. (1998), Principles of Population Genetics

(3rd ed.), Sunderland, MA: Sinauer.

Li, W.-H. (1997), Molecular Evolution, Sunderland, MA: Sinauer.

Nonlinear Estimation and Classification.

David D. Denison, Mark H. Hansen, Christopher C. Holmes, Bani Mallick, and Bin Yu (Eds.). New York: Springer-Verlag, 2003.

ISBN 0-387-95471-6. vii + 474 pp. $69.95 (P).

As an addition to Springer's series of Lecture Notes in Statistics, Nonlinear

Estimation and Classification is a compilation of 31 articles on recent develop ments on estimation and classification with nonlinear methods. The book grew out of a 2-week-long workshop of the same title held in spring of 2001 at the

Mathematical Sciences Research Institute in Berkeley, California. The organiz ers of the conference set a goal of promoting communication between several

disciplines that address issues in nonlinear estimation and classification, but of

ten with different emphases, terminology, and perspectives. Such disciplines in

clude machine learning, signal processing, image analysis, information theory,

optimization, applied mathematics, and of course statistics. The main charac

teristics and challenges that make these research topics exciting include high

dimensionality of the input variables, complex structure of the data, and the

need to process massive amounts of data. The traditional linear statistical tools

are generally incapable of handling such situations.

The articles are grouped as long papers and short papers. The 11 long pa

pers (with the average length of 20 pages) are from invited talks given by some

leading researchers, and the remaining 20 papers (with an average length of

11 pages) are from contributed talks that also address important questions in

the area. The papers are a mixture of review articles and presentations of new

research results.

Some of the long papers concentrate on specific areas of application or on

specific statistical issues; there are also review articles on general methodolo

gies. The papers that focus on a single area of application include natural image

modeling by wavelets (Choi and Baraniuk), scene labeling via a coarse-to

fine strategy (Geman), environmental monitoring using satellite images (Kiiveri et al.), freeway traffic-flow modeling (Bickel et al.), and internet traffic model

ing (Cao et al.). These works introduce complex modeling issues in real-world

applications. Marineantu and Dietterich compare two methods of estimating the conditional probability function in classification based on trees. Gray ad

dresses the problem of clustering Gaussian mixtures with quantization based

on information-theoretic tools.

Huang and Stone review extended linear modeling with tensor-product

splines, a general framework proposed and studied by Stone and collabora

tors. Theoretical, methodological, and computational aspects are touched on,

561

with a special emphasis on free-knot cases. Two exciting methods of clas

sification developed outside of statistics are reviewed, namely support-vector machines (SVMs) and boosting. Wahba et al. offer insight on SVMs from a tra

ditional statistical viewpoint, review adaptive tuning of SVMs, and also address

nonstandard SVM problems. Schapire, who pioneered the now-popular boost

ing method for improving a weak classifier, reviews its application in machine

learning. He covers the basics, error bounds, connections to game theory, lin

ear programming, and logistic regression. Some demonstrations are also given.

Mukherjee et al. examine the familiar regularization approach to regression,

especially from a stability standpoint, and derive useful error bounds.

Overall, I find the long papers quite readable, informative, and sufficiently

explained to provide a reasonably clear picture of the topics. One helpful as

pect is that efforts are made to explain things from the perspective of different

disciplines and connections (e.g., statistical views on boosting). The short papers cover a diverse range of issues, including risk bounds for

estimating a regression function or hazard function based on trees (or combin

ing trees) or neural networks of different kinds; considerations of prior assign ment for Bayesian hierarchical models or nonparametrics; data compression for remote sensing data; clustering in nonstandard contexts; wavelets or related

methods for categorical data, nonequispaced data, or high-frequency financial

data; confidence intervals for spline density estimation; longitudinal or growth curve modeling using mixed-effects MARS; clustering methods in bioinfor

matics; adaptive kernels for SVMs; ANOVA DDP models for a set of related

random distributions; logic regression using Boolean combinations of binary

covariates; and instability in nonlinear estimation and classification. Most of

these articles report some new results, but usually in a more or less sketchy way

(presumably due to space limitation), whereas a few others are mainly review

papers on a specific topic. Theoretical, methodological, and computational is

sues are considered.

Two notable features of the book are its wide coverage in content and its

self-contained chapters. This is, of course, not surprising given the background of the book. A consequence is that one can read the papers (called chapters in

the book) in any order according to personal interest.

The book's strength lies in the joint depiction by the active researchers

of the recent research works on nonlinear estimation and classification, some

from fields other than statistics. The reviews of the important (relatively) new

methodologies, some classes of applications, and computational aspects should

be quite accessible to a wide range of readers. The book is certainly worth read

ing for a researcher seriously interested in understanding the state-of-the-art

methods and potential applications in high-dimensional estimation and classi

fication. I also think that the book can serve well for teaching a topics course

on modern estimation and classification in which advanced graduate students

select a topic for study and presentation. Of course, this book is not meant to be exhaustive or a perfect representation

of the recent and the current works under the theme of nonlinear estimation and

classification, nor is it sufficiently detailed for an in-depth understanding of the

technical issues. Indeed, in most cases the resolution levels of the materials are

relatively low, and the reader has the opportunity to use the references provided in the articles for further reading.

Nonlinear estimation and classification is an important and rapidly growing research area that has benefited from several disciplines. From this book, it is

clear that many challenges and opportunities remain for future research.

Yuhong Yang

Iowa State University

Nonparametric Goodness-of-Fit Testing Under Gaussian Models.

Yu I. INGSTER and Irina A. SUSLINA. New York: Springer-Verlag. 2003.

ISBN 0-387-95531-3. xiv + 452 pp. $98.00 (P).

This book concerns nonparametric goodness-of-fit testing problems. The

chosen approach is the minimax asymptotic approach (small noise level or

large sample size). The authors explain the choice of the minimax tests

by first studying the classical goodness-of-fit tests (Kolmogorov-Smirnov, Cram?r-von Mises). They also highlight the similarities and differences be

tween estimation and testing. The most modern aspects are studied; including

adaptive problems, multichannel detection, and composite null hypotheses. The

This content downloaded from 195.34.79.253 on Sat, 14 Jun 2014 13:27:00 PMAll use subject to JSTOR Terms and Conditions

562

book is self-contained, and the bibliography is very rich and in fact provides a comprehensive listing of references about minimax testing (something that

heretofore had been missing from the field). The authors explain the minimax approach and its connection with the

Bayesian approach. There are two kind of problems: the sharp asymptotic prob lem and the simpler problem about distinguishability and undistinguishability conditions.

To get the best out of this book, the reader should be familiar with basic func

tional analysis, wavelet theory, and optimization for extreme problems. There

are only a few typographical errors, and they can easily be corrected by the

reader. This book is intended for graduate students and researchers. Parts of

this book can serve as starting points for a course at a graduate level. It is highly recommended to anyone who wants an introduction to hypothesis testing from

the minimax approach?yet it is only a starting point, as Gaussian models are

studied exclusively. For other models, the reader will have to get articles or wait

for another book on density or spectral density models, maybe that by Ingster and Suslina.

The authors begin with the distinguishability problem. Their purpose is to in

troduce the finite-dimensional and infinite-dimensional problems. They clearly

explain the need for smoothness assumptions, especially the importance of the

norm to define the removed neighborhood. They make the connection between

sequence spaces (power norms and Besov norms are considered) and functional

spaces (Besov and Sobolev spaces) through the wavelet theory. Differences between estimation and hypothesis testing are indicated. The

testing problem is important because the plug-in estimation test does not pro vide a good rate of convergence in nonclassical results. Conditions for distin

guishability and undistinguishability are given for ellipsoids. The concept of

semiorthogonal priors is also introduced.

The tools introduced for the study of the lower bound are very important and

are applied systematically in the case of power norm, Besov norm, and positive alternatives. An interesting point that the authors make is about the Besov norm,

the parameters can lead to trivial results.

Next, the sharp asymptotics problem is considered, and the links between

the upper bound and the lower bound are clearly explained. This leads to a very

general problem of optimization. This approach is very demanding, but gives a deep understanding of the results. A general approach is given to justify the

choice of two- and three-point priors. Sharp asymptotics lead to an optimization

problem on the space of probability measures.

In the infinite-dimensional model with the level noise fading, three types of

results are obtained: trivial, asymptotics of degenerate types, and sharp asymp totics of Gaussian types.

The last part of the book is devoted to more specialized and modern topics. The problem of adaptation, introduced in testing by Spokoiny in 1996, is con

sidered here in the case of the power norm and Besov norm. The second type of

result involves high-dimensional signal detection. Here new kinds of asymptot ics with infinite-divisible distributions are obtained. This problem is related to

multichannel signal detection. The third modern topic is the problem of testing a composite null hypothesis, especially a hypothesis that is close to a simple

hypothesis. The rates of testing are the same as in the simple null hypothesis

counterparts. Yet, the size of the composite null hypothesis depends heavily on

the distance that measures this size (Lp norm).

Christophe Pouet

Universit? de Provence

REFERENCES

Lepski, O. V, Nemirovsky, A., and Spokoiny, V G. (1999), "On Estimation

of the Lr Norm of a Regression Function," Probability Theory and Related

Fields, 113, 221-253.

Spokoiny, V. G. (1996), "Adaptive Hypothesis Testing Using Wavelets," The

Annals of Statistics, 24, 2477-2498.

Applied Probability.

Kenneth Lange. New York: Springer-Verlag, 2003. ISBN 0-387

00425-4. xii + 300 pp. $79.95 (H).

Professor Lange has presented us with a graduate-level text in probability. In his Preface, he worries that the pursuit of mathematical rigor discourages students of science, particularly of biology, from learning the powerful tools

Book Reviews

that modern probability theory puts at their disposal. This book is an attempt to remedy that. The first chapter is a brief, quick introduction to mathemati

cal probability, with almost no proofs. I believe this chapter should constitute

a warning to students of the sorts of things they should already have seen in

their education, that they may be called on to remember and use in what fol

lows. An introductory graduate, or perhaps upper-level undergraduate, course

in mathematical probability should usually suffice.

The rest of the book is a series of chapters that introduce probabilistic tools,

with many illustrative applications, to mathematical biology (particularly ge

netics) and other branches of mathematics. The author's professional interests

certainly guided his choices, but the selection of methods and areas of applica tion is nevertheless so broad that readers with other specialties will likely find

their curiosity piqued. This manner of development meant that I often would have chosen alterna

tive approaches to the applications; but this seems beside the point. The author

wanted to show the breadth and power of the tools. Several times I was pleas

antly surprised to learn that an application could be tackled in an unfamiliar

way. Because the book is comparatively short, it can by no means be a survey

of all the important tools of modern applied probability. For example, little is

given of asymptotic approximation theory, and comparatively little of stochas

tic processes. Nor are the tools that do appear studied to great depth. But the

ones that are covered seem well chosen and are well explained. More than once

I found informative discussions of state-of-the-art techniques that I had recently encountered in the research literature and wished to learn more about. The book

is rich with pointers to an excellent bibliography for those wishing to pursue

subjects further.

Chapter 2 covers devices for finding expectations, including familiar ones

like indicators and generating functions and less-familiar tricks using condi

tioning. Chapter 3 is an entertainingly idiosyncratic chapter on convexity, with

some standard connections to inequalities, and less familiar ones to optimiza tion problems.

Chapters 4 and 5 consider combinatorics, at an intermediate level beyond that known to every student of probability. The techniques exploited include

inclusion-exclusion and the pigeonhole principle. Applications include an in

teresting introduction to probabilistic techniques for evaluating algorithms.

Chapter 6 introduces Poisson processes, with applications to such fields as

tomography. There are several illustrations of what the author calls Poissoniza

tion; let an integer parameter be a Poisson random variable, then condition on

its value. Later, Chapter 12 introduces some rather modern techniques in Pois

son approximation, such as the Chen-Stein bounds.

Chapters 7 and 8 introduce the classic topics of discrete and continuous

Markov chains. Chapter 9 provides a nice introduction to branching processes. All of these chapters present a number of biological applications, such as to

genetics and epidemiology.

Chapter 10, on martingales, will no doubt please statisticians for its applica tion to sequential testing, but will equally certainly frustrate specialists for its

brevity. Chapter 11, on diffusion processes, is surprisingly rich in theory and

applications even at this fairly elementary level. Chapter 13 is an unexpected

chapter, dealing with probabilistic techniques in number theory. It climaxes

with a proof of the prime number theorem, of all things. Professor Lange's book is certainly a pleasure to read, and it is written in a

clear style using standard probabilistic notation. Each chapter ends with exer

cises ranging from moderate to challenging in difficulty. These exercises serve

to fill in mathematical details, elaborate on the applications discussed, and sug

gest further applications. I expect to keep this book handy, because my interests require that I be able

to resurrect knowledge of some of the many techniques found within. The au

thor suggests that the book may be useful in teaching graduate students in the

sciences the probability that they need for their research. In my experience, very few graduate students in the sciences are ready for the mathematical level re

quired here. Rather, the book perhaps could be used as the text for the second

part of a serious graduate course in probability for an applied probability, statis

tics, or mathematics program. The first part of the course would be taught from

a good mathematical probability textbook, which this book clearly presupposes.

George R. TERRELL

Virginia Polytechnic Institute and State University

This content downloaded from 195.34.79.253 on Sat, 14 Jun 2014 13:27:00 PMAll use subject to JSTOR Terms and Conditions