Infinitely Divisible Point Processes. by K. Matthes; J. Kerstan; J. MeckeReview by: Alan F. KarrJournal of the American Statistical Association, Vol. 75, No. 371 (Sep., 1980), pp. 750-751Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2287686 .

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750 Journal of the American Statistical Association, September 1980

Neither book is likely to be of use in teaching formal statistics courses, although either one could serve as auxiliary reading on the policy impacts of statistical analysis or on the social respon- sibility of the statistician. Both books will be important for anyone concerned with public policy analysis or the logic of Marxist and other nontraditional criticism of statistical orthodoxy.

The reviewer has benefited from comments by W.H. Kruskal on the Irvine et al. book, which have contributed to the prepara- tion of this review.

MACK SHELLEY Iowa State University

Survival Probabilities: The Goal of Risk Theory. Hilary L. Seal. New York: John Wiley & Sons, 1978. x + 103 pp. $24.50.

The word risk-along with such expatiations as risk analysis, risk assessment, and perceptions of risk-currently occupies a great deal of governmental activity. Such diverse concerns as dam con- struction, drug certification, and nuclear licensing employ the language and concepts developing around the quantification of risk; see, for example, the books by Lowrance (1976) and Rowe (1977). Thus it was that Seal's Survival Probabilities, bearing the subtitle

The Goal of Risk Theory, was eagerly awaited by this reviewer. And thus it was that the dust jacket's message reduced that eagerness by stating:

This book is concerned with numerical probabilities of ruin for realistic models for nonlife insurance companies and the associated computer pro- grams. It deals with the probability of an insurance company surviving a limited time period during which the annual number of claims is expected to be reasonably constant but which could increase significantly. This can be calculated both easily and accurately.

But a basic reviewing truism reminds us not to criticize a work for what it is not. Rather, our job is to examine the work for what it is and how well it is executed for its intended purpose.

In five chapters, totaling 63 pages, the reader is swept from a "historical introduction" through a "vale to queueing techniques" and onto "a computational accessory-the Laplace transform" to computing "the probability of t-years survival" of the firm with "approximations and controls" for computer-assisted evaluation of the required functions.

A 33-page appendix is devoted to ". . . eight FORTRAN pro- grams that produced most of the numerical results in the preceding chapters." (This reviewer, at least, would prefer that the word "most" had not appeared in that quoted phrase, but this may be Seal's way of leaving something for the reader.) The program listings are cleanly produced; there should be little problem in transcribing them should a user fail to locate tape or card versions from some other source.

Well over 50 citations appear in the bibliography; they span some 150 years of work and thus provide the reader with a useful perspective of the subject. The earliest, which appeared in 1835, is by T. Barrois and titled "Essai sur l'application du calcul des probabilit6s aux assurances contre l'incendie." The latest, pub- lished in 1978, is by Seal himself and titled "From Aggregate Claims Distribution to Probability of Ruin." No matter how shaky our French may be-or our English, for that matter-all of us should appreciate the nature of the bracketing of Seal's subject performed by these two papers' titles.

This is not only a book for those who like/need this type of book-it is a book for anyone who delves into the economics of insurance.

But there is even better news for the scholars and practitioners who have been characterized in a popular essay by Epps (1979) as "death bookies." Seal's models and analyses should be im- mediately applicable to physical risk issues. It is devoutly to be wished that proper attention be paid to Seal's work: It's worth it.

ROGER H. MOORE Department of Commerce,

Bureau of the Census

REFERENCES

Epps, Garrett (1979), "They Bet Your Life," The Washington Po8t Magazine, November 11, 38-46.

Lowrance, William W. (1976), Of Acceptable Risk, Los Altos, Calif.: William Kaufman.

Rowe, William D. (1977), An Anatomy of Risk, New York: John Wiley & Sons.

Infinitely Divisible Point Processes. K. Matthes, J. Kerstan, and J. Mecke. New York: John Wiley & Sons, 1978. xii + 532 pp. $42.50.

Point processes have for some time been an active area of re- search in probability theory and are becoming increasingly im- portant in statistics and applications, particularly in the engineering sciences. This book is a detailed and meticulous exposition of the "distribution theory" of point processes and is based upon the authors' Unbegrenzt Teilbare Punktprozesse (1974). Unfortunately, the style and notation render the material inaccessible to all but specialists.

A point process M on a set G (viewed as a random distribution of points in 0, with M(A) the number of points in A C G) is said to be infinitely divisible if for each n it is equal in distribution to the superposition M1 + . . . + Mn of n independent, identically distributed point processes M1, . .., Mn. The infinitely divisible point processes, of which the Poisson processes on R are the most familiar examples, form the main, but not the only, subject matter of the book.

The book is most impressive in terms of completeness and detail. A brief description of the contents of the more important chapters follows.

Chapter 1 (116 p.) is an excellent introduction to the measure- theoretic and topological bases of point process theory, and presents such concepts as intensity measures, Poisson processes, the Poisson mixtures UE, mixed Poisson processes, the thinning operation, and marked point processes.

Chapter 2 (29 p.) contains mainly an important characterization (Theorem 2.1.10) of infinitely divisible point processes as Poisson mixtures E w.

Chapter 4 (52 p.) presents the fundamental characterization (Theorem 4.5.3) of infinitely divisible point processes as admitting a clustering representation in the following manner: Begin with a suitable Poisson point process of "cluster origins" and from these points, using a suitable transition kernel, construct a set of "secondary points," which then constitute the infinitely divisible point process.

Chapters 5 (32 p.), 8 (27 p.), and 9 (43 p.) deal with Campbell measures, Palm measures, and Palm distributions. The Campbell measure is a generalization of the intensity measure, while the Palm distribution is a mixture of probability measures that can be regarded as conditional laws of the point process, given that a point is at a prescribed location. These chapters contain various characterizations of infinitely divisible point processes, the clas- sical Palm-Khinchin formula, and a discussion of stationary renewal processes.

Chapter 7 (22 p.) is a very nice treatment of Cox processes (--doubly stochastic Poisson processes), including the principal characterization in terms of thinning (Theorem 7.2.8).

Chapter 10 (38 p.) contains the "generalized Palm-Khinchin" Theorem (10.3.9) and related results characterizing infinitely divisible point processes as limits of row-wise superpositions of infinitesimal triangular arrays of point processes; some elementary results of this form appear in Chapter 1.

The remaining chapters deal with weak convergence theory for point processes, stationary point processes on R', homogeneous cluster fields and spatially homogeneous branching processes. There is a bibliography of 235 entries that, relative to the subject matter of the book, is quite complete through 1976. References and attributions are scattered throughout the text rather than (as the reviewer prefers) being centralized at the ends of chapters or the end of the book.

To the worker in point processes, this book, despite some of the flaws noted here, will be indispensable. The only other uni- fied treatment of (some of) this material, Kallenberg's (1976) Random Measures, is less complete (but also much less difficult). Infinitely Divisible Point Processes is a valuable research tool and could (albeit not without difficulty) be used in advanced seminars.

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Book Reviews 751

To nonexperts, including most students, the book is a closed one: Through an abstruse style and blizzard of notation the authors have made most of the wealth of material in the book virtually inaccessible to nonspecialists. The notation, in particular, is a nightmare. Associated with the distribution P of a point process are (to mention only a few) objects denoted by P, P2, P2, PI, J(P), PO) Po, P, (P)?, (Po)', and (P). Keeping them straight was beyond the typesetters at times and will be equally hard for many readers. Arguments used, while correct, are frequently ponderous and hard to follow; motivation and interpretation are almost nonexistent. The nonexpert might better begin with Kallenberg's book or Snyder's Random Point Processes (1975).

Finally, the reviewer concurs with the authors' regret at their not including recent work involving Gibbs distributions and con- ditional intensities and wishes to point out that the book omits entirely the sample path theory of point processes typified by the "martingale approach."

ALAN F. KARR T'he Johns Hopkins University

REFERENCES

Matthes, K., Kerstan, J., and Mecke, J. (1974), Unbegrenzt Teilbare Punktprozesse, Berlin: Akademie-Verlag.

Kallenberg, 0. (1976), Random Measures, Berlin: Akademie-Verlag. Snyder, D. (1975), Random Point Processes, New York: John Wiley & Sons.

Selecting and Ordering Populations: A New Statistical Methodology.

Jean Dickinson Gibbons, Ingram Olkin, and Milton Sobel. New York: John Wiley & Sons, 1978. xxi + 569 pp. $24.95.

Methodologies for selecting and ordering (ranking) populations were developed to provide experimenters with meaningful statistical formulations of real-life problems that often had been inappro- priately posed and analyzed as tests of homogeneity. It is now some 25 years since the first published research papers appeared proposing statistical approaches for designing and analyzing ex- periments that were concerned with such selection problems. Since that time upwards of 500 papers that further developed, generalized, and integrated the theory underlying these methodologies have appeared in a large number of journals. Until now, no book has been written for use by applied statisticians that deals exclusively with these techniques. With their present text, Gibbons, Olkin, and Sobel (GOS hereinafter) have undertaken the formidable mission of attempting to digest, sort out, and present the most relevant of these methodologies. It is indeed the first of its kind, and the profession is indebted to the authors for their pioneering effort. In general, they have done a good job, and we recommend the text for both applied and theoretical statisticians (but mostly for the former). However, applied statisticians and teachers of "selection and ordering" courses should especially be aware of a number of errors (some serious) and several unfortunate omissions. Some of these that this reviewer has noted are detailed in Sec- tions 2 and 3. Teachers and statistical researchers may also wish to consult the new comprehensive text on this subject by Gupta and Panchapakesan (1979).

1. INTRODUCTION

An overview of the text follows: Chapter 1 sets the stage for the remainder of the book. It discusses the philosophy of selecting and ordering populations, poses possible ranking and selection goals, and introduces the concept of a "measure of distance" between populations and the probability requirement (stated in terms of this measure) for the so-called "indifference-zone" ap- proach. The concept of a "least-favorable" configuration of the population parameters associated with the probability requirement and the given selection procedure is introduced, and its role in sample-size selection is discussed. Subsequent chapters specialize these general notions to specific probability distributions and goals. Chapter 2 deals with the problem of selecting the normal popula- tion with the largest population mean when the common variance is known, and Chapter 3 deals with the corresponding problem when the variances are (a) known but unequal, (b) common but unknown, or (c) completely unknown. Chapter 3 also considers

other problems for normal means, including that of estimating the largest population mean. The problems of selecting the Ber- noulli population with the largest single-trial "success" proba- bility, the normal population with the smallest variance, and the multinomial category with the largest (smallest) "event" proba- bility are considered in Chapters 4, 5, and 6, respectively. Chapter 7 deals mainly with a nonparametric procedure for selecting the population with the largest qth quantile. In Chapter 8 the authors offer new (hitherto unpublished) results concerning selection pro- cedures for a design with paired comparisons. The problem of selecting the normal population with the "best" regression values is posed in Chapter 9. Chapter 10 studies selection procedures for comparisons wrt a control or a standard for the normal means and the normal variances problem. Chapters 11 and 12 consider different selection goals, that is, selecting the t (1 _ t ? k - 1) best of k populations (without regard to order) for the normal means problem in Chapter 11 and the complete ordering of k populations for the normal means and for the normal variances problems in Chapter 12. Almost all of Chapters 1-12, part of Chapter 14, and all of Chapter 15 deal with selection from the viewpoint of the indifference-zone approach; only Chapter 13 and a portion of Chapter 14 deal with selection using the subset ap- proach. Chapter 13 is concerned with the problem of selecting a subset of the k _ 2 normal populations containing the popula- tion with the largest population mean when the common variance is known, or a subset of the k _ 2 Bernoulli populations con- taining the population with the largest single-trial "success" proba- bility. Also considered in Chapter 13 is the problem of selecting a subset of the k populations containing all populations at least as good as a standard or control-both for means of normal popu- lations when the common variance is known, and for single-trial "success" probabilities of Bernoulli populations. New (hitherto unpublished) results concerning an application of subset selection to the scoring of tests are also given. Chapter 14 deals with the application of the indifference-zone and subset approaches to selection problems concerning Gamma populations. Finally, Chap- ter 15 considers selection procedures for multivariate populations- selecting the best component from a single p-variate distribution, or selecting the best of k _ 2 p-variate distributions.

Each chapter contains several "concrete illustrations of applica- tions to indicate that the model under consideration is a prototype for many practical problems in diverse areas of experimentation and/or general investigation." A large number of worked examples are given throughout the text, indicating the vast range of ap- plicability of the procedures. Problems are provided at the end of each chapter, many involving real (with references) or artificial data sets, which the student can use to test his understanding of the material covered in that chapter.

An outstanding feature of the book is the set of 48 tables (ap- proximately 140 pages) and four figures given in the Appendix; these are required to implement the procedures described in the text, or to study their performance. The tables were gathered and abstracted from the many statistical journals in which they first appeared (several tables are new or were recomputed) and are presented in a clear, uniform, easy-to-read format. Detailed inter- polation instructions are given. The book also contains an "ap- plications" and a "theory" bibliography.

This reviewer particularly enjoyed reading Chapter 1, which is well written and conveys clearly the spirit of the selecting and ordering point of view. Chapter 6, on the multinomial distribu- tion, is for the most part a fine one, and the figures in barycentric coordinates contribute substantially to an understanding of the preference and indifference zones and least-favorable configuration for selecting the category with the largest "event" probability. (Equivalent figures for the problem of selecting the category with the smallest "event" probability would also have been helpful.)

Of particular importance is the stress that the authors place on the confidence statement that the experimenter can make after the experiment has been completed and a population selected (p. 21, par. 2). The interplay of the confidence coefficient, width of confidence interval, and sample size is described, and it is shown how these can be used to provide a rational basis for the choice of sample size for single-stage procedures. (It is not pointed out, however, that analogous confidence statements can be made for multistage or sequential experiments, but that for these latter ex- periments the "design" aspects of the experiment are much more complicated.)

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