Patrick Billingsley Probability And Measure Pdf
Patrick Billingsley 4 May 1999 Seven Lectures on “Weak Convergence of Probability Measures,” delivered at the Edinburgh Mathematical Society’s quadrennial.
. Probability and Measure. By Patrick Billingsley Review by: Richard C. Bradley Journal of the American Statistical Association, Vol.
399 (Sep., 1987), pp. 946-947 Published by: American Statistical Association Stable URL:. Accessed: 15:48 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of 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 Journal of the American Statistical Association. This content downloaded from 62.122.76.48 on Sun, 15 Jun 2014 15:48:04 PM All use subject to JSTOR Terms and Conditions. 946 Journal of the American Statistical Association, September 1987 Beyond ANOVA, Basics of Applied Statistics. New York: John Wiley, 1986. Xiii + 317 pp.
I looked forward to reading this book because of the opening remarks of the preface: These are the confessions of a practicing statistician. They expose to public view what I am likely to do with a set of data. I may therefore live to regret setting pencil to paper. Yet there does not seem to be a book that tells a student how to attack a set of data.
V) The author sincerely wishes to share practical lessons from many years of experience. The style continues in this friendly manner throughout the book. The organization is similar for each of the seven main problem areas considered: one-sample problems, two-sample problems, one-way clas- sification, two-way classification, regression (including errors-in-variables and calibration), ratios, and variances. Miller starts with the normal the- ory formulation and then systematically dissects the procedure as as- sumptions (normality, equal variances, independence) are relaxed. Where possible, he provides adjustments or alternative well-behaved procedures.
In this respect, the book would be extremely valuable to recent M.S. Statistics graduates in organizing their thinking for applications. The book should provide such an audience a healthy skepticism for ap- plying procedures in an automaton fashion. In reading this book, it becomes evident that the author's hard-won strategies emerged from very careful reading of The Annals of Statistics, Biometrika, JASA, and the Journal of the Royal Statistical Society (JRSS), Series B.
He is particularly attuned to methods giving guidelines for back- of-the-envelope calculations. This view of applied statistics, however, does not seem particularly modern in view of ready access to computers and statistical packages.
Electronic calculators seem to be the author's most sophisticated crutch, whereas computers are a last resort for prob- lems requiring immense amounts of calculations. This view is dated, particularly for the one-sample problem. Given a set of numbers (even just a few), the first thing I do is create a data file and subject it to computationally reliable statistical software compare this with the ob- solete formulas (5.7), p. I probably will not use all of the generated output, but at least the computational drudgery is eliminated and I can pick and choose what to highlight for a client.
In a similar vein, the author seems to place simulation studies in the second-string role of confirming theoretical conjectures. This is too bad because miniature simulation studies can often be geared to particular data analysis problems to easily assess assumptions. The preceding objections could be countered by as- serting an overreliance on computers-so be it. It seems amazing that a book with 'Applied Statistics' in the title would have only seven references to Technometrics and none to Applied Statistics (JRSS, Ser.
As a case in point, Chapter 7 could have been updated by considering the Technometrics review article by Conover, Johnson, and Johnson (1981). One other complaint that I have with the book is the selection of font. The text was prepared using the increasingly popular TEX. The font is not very attractive, and the symbols are hard to make out at times. In Chapter 5, multiple circumflexes are used without explanation. I assume that these are unintentional. I found the book useful for collecting information on various topics in a neatly organized fashion.
Perhaps because of the intended audience and scope, it seems that these basic problem areas (one-sample, two- sample, etc.) are still a little too clean. Two-sample problems with a conclusion that the populations look the same (even taking into account the various assumptions) generally appear suspicious. Further probing commonly reveals that experimental conditions have not been quite the same, so that auxiliary information should be included. I cannot help but wonder if Miller was joking when he said: The sign test is a marvelous device for hurriedly getting the client out of your office. He or she will be happy because the data have received an official stamp of statistical significance, and you will be happy because you can get back to your own research.
21) Perhaps it would have been better not to share this honesty with the recent M.S. Or Ph.D reader, who might mimic this 'data-side' manner. Although I disagree in places with what Miller says, I admit that I like how he says it. Miller's writing style is downright entertaining. I prefer informal discussions to drive home key points-Miller does this beau- tifully. A few of my favorites follow: 'It cannot be denied that many journal editors and investigators use P c.05 as a yardstick for the publishability of a result.' 3) 'With unequal slopes, any bizarre collection of lines is possible with irregular criss-crossing like the game of 'Pick Up Sticks'.'
18) 'God has not decreed that all regressions should be linear.' 193) 'Outliers are a disaster story.' 199) In summary, this is a very original book. Its style and organization are particularly well suited for preparing novice data analysts for the world of real, rather than theoretical, statistics. The computational aspects will probably outdate the book sooner than it deserves, and the font could be prettier.
Patrick Billingsley
The book is certainly worth reading, however, and probably worth owning as an occasional reference book for the seven basic problem areas covered. I reviewed this volume during a business trip to Boston.
As compe- tition, I also brought along Blue Highways and Citizen Hughes. When was the last time you finished a statistics book before a best-seller and an expose? JOHNSON Los Alamos National Laboratory REFERENCE Conover, W. J., Johnson, M. E., and Johnson, M. (1981), 'A Comparative Study of Tests for Homogeneity of Variances, With Applications to the Outer Continental Shelf Bidding Data,' Technometrics, 23, 351-361. Probability and Measure (2nd ed.).
Patrick Billingsley. New York: John Wiley, 1986. Xii + 622 pp.
Billingsley gives a rigorous and fairly standard treatment of probability theory, with much attention to measure theory, some discussion of topics arising in statistics (e.g., the Glivenko-Cantelli theorem, minimum-var- iance estimation, likelihood ratios, and Bayes estimation), and a solid treatment of basic stochastic processes e.g., Markov chains, Poisson processes, (sub)martingales and reversed martingales, and Brownian mo- tion. Billingsley also discusses in detail several other topics generally not covered in a standard first-year course, including random permu- tations, Skorohod embedding, the weak invariance principle, extremal distributions, the central limit theorem (CLT) for strictly stationary se- quences satisfying the 'strong mixing' condition, Hausdorff measures, and topics in number theory that can be treated with probabilistic meth- ods.
In an early chapter on simple random variables, there is a series of exercises that leads the reader tantalizingly close to a proof of the prime number theorem but unfortunately stops just short. The remainder of this review will be addressed primarily to the book's strengths and weak- nesses in its treatment of the standard topics that one might wish to cover in a first-year course. Billingsley gives an adequate treatment of all of the 'right' topics for such a course.
They are arranged in a good order for pedagogical pur- poses-, with the level of sophistication starting out low and increasing slowly as one goes along, culminating with a rigorous treatment of Kol- mogorov's existence theorem and Brownian motion. The book has a lot of material, including many examples. In places it is a bit dense or terse. The material is carefully organized, however. The level of mathematics is suitable for a serious, rigorous, no-nonsense first- year graduate-level course. Students who are strong enough potentially to get a Ph.D.
Degree in mathematics should be able to use the book effectively as a textbook. Students who are not so strong will have some difficulty with it. The treatment of each topic is well thought out. Technical details are given meticulous care and preparation and are generally handled cleanly, without much 'sweat.'
There do not seem to be many errors and no serious ones of which I am aware. One appreciates the care with which the book is written, especially in those spots where the technical details are by nature tricky or delicate. A wide variety of exercises covers essentially every topic treated in the text. Among those exercises is an ample supply suitable for a standard course.
Probability And Measure Pdf
Billingsley gives an extensive, self-contained treatment of measure and integration theory (henceforth called just 'measure theory'). He derives from scratch all of the results in measure theory that are needed for the study of probability theory at this level.
The only prerequisites for the book are rigorous calculus (including power series), elementary set the- ory, and some matrix theory. Each piece of measure theory is introduced at the place where it is first needed. For example, the Radon-Nikodym derivative is not estab- lished until late in the book, just before the treatment of conditional probability/expectation given a cr field.
Probabilistic topics that require only very little measure theory are treated early-for example, the Borel- Cantelli lemma, the strong law of large numbers for simple random This content downloaded from 62.122.76.48 on Sun, 15 Jun 2014 15:48:04 PM All use subject to JSTOR Terms and Conditions. Book Reviews 947 variables, and the asymptotic behavior of countable-state Markov chains. Early in the book one gets a good idea of what probability theory is all about and of why some further measure theory might be desirable for the broader study of probability. The repeated sandwiching of measure theory and probability is in spirit similar to that used by Adams and Guillemin (1986), which is a less sophisticated, less extensive textbook intended for proficient undergraduate mathematics majors. My only reservation concerning Billingsley's treatment of measure theory is his early use of 7r systems, A systems, and Dynkin's 7r-2 theorem.
This allows a quick, efficient development of several results in measure theory. In such places however, other tools involving just fields and ar fields (e.g., the monotone class theorem) will suffice without too much extra work. For students who have had little or no measure theory (the book is intended partly for such students), it would perhaps be easier to start out by working with just fields and ar fields until one is quite familiar with them and then to become acquainted with 7( systems, A systems, and the 7r-A theorem later. Billingsley's treatment of weak convergence in multidimensional Eu- clidean space-including, for example, the Cramer-Wold techniques, the multivariate normal distributions, and the multivariate CLT-is nicely organized if a little terse.
This topic was not treated as thoroughly by Chung (1974). Billingsley does not discuss weak convergence in more general separable Hilbert spaces or Polish spaces as Laha and Rohatgi (1979) did, but the treatment in multidimensional Euclidean space is arranged so as to facilitate extension to more general spaces. Billingsley's treatment of stochastic processes is, on the whole, slightly more extensive than Chung's (1974) and far more so than that by Laha and Rohatgi (1979), who treated martingale theory well but did not seem to mention Markov chains, Poisson processes, or Brownian motion.
For martingale theory and for nondenumerable Markov chains, the treatment by Chung (1974) seems slightly more well-rounded, with a slightly better finishing touch than that by Billingsley. If one wishes to teach a first- year, graduate-level probability course that puts heavy emphasis on sto- chastic processes (say with a rigorous treatment of the Birkhoff ergodic theorem, random harmonic analysis, prediction theory, and a broad class of Markov processes), there are more suitable textbooks, such as Ro- senblatt (1974). Billingsley proves the CLT (both one-dimensional and multidimen- sional) only via characteristic functions. For a more direct insight into the CLT than via transforms, one has to look elsewhere. For example, in one dimension, Chung (1974) and Rosenblatt (1974) each gave a direct proof of the CLT (and another proof using characteristic functions).
Feller (1968) proved the local limit theorem for binomial random vari- ables (DeMoivre's result). These various criticisms of Billingsley are very minor. Such differences of taste are easily incorporated into one's lectures. For a standard and rigorous first-year course in probability theory, with some attention to measure theory, statistics, and stochastic processes, Billingsley has writ- ten what is clearly one of the best textbooks. I am not aware of a better one for this purpose. Overall, the 1986 edition seems to be a slight improvement on the 1979 edition.
There are more examples and more exercises. A few proofs have been rewritten. An appendix has been inserted, giving a discussion of numerous mathematical topics and techniques that are used in the text-for example, the diagonalization method, basic properties of con- vex functions, and basic properties of power series.
The section on countable-state Markov chains has been slightly extended and improved. In the section on martingales, stopping times are used earlier (e.g., in the proof of the upcrossing inequality), a CLT for martingales is proved, and a discussion of exchangeable random variables that was in the 1979 edition has been deleted. The classical proof of the strong law of large numbers under finite absolute first moments has been replaced by Ete- madi's well-known recent proof. Kolmogorov's maximal inequality, how- ever, has been retained in the new edition. The treatment of Poisson processes has been extended, and a brief discussion of the Fourier series for probability measures supported on 0, 27r has been added. BRADLEY Indiana University at Bloomington REFERENCES Adams, M., and Guillemin, V. (1986), Measure Theory and Probability, Belmont, CA: Wadsworth.
(1974), A Course in Probability Theory (2nd ed.), New York: Ac- ademic Press. (1968), An Introduction to Probability Theory and Its Applications (Vol. 1, 3rd ed.), New York: John Wiley. G., and Rohatgi, V. (1979), Probability Theory, New York: John Wiley. Rosenblatt, M. (1974), Random Processes (2nd ed.), New York: Springer-Verlag.
Probability and Random Variables. Chichester, U.K.: Ellis Horwood, 1986. This book is appropriate for a one-semester, calculus-based under- graduate course in probability theory.
The book's intent is to present the topic of 'probability with statistics in mind' (p. Beaumont accomplishes this by using many examples and problems illustrating how the techniques are useful in real situations from various disciplines. This feature, the strength of the book, separates it from competitors that present probability through a series of uninteresting experiments in- volving coins, urns, and so forth; it thereby encourages the student to take a second course in statistics.
Beaumont covers the traditional topics of an undergraduate course in probability theory. Many readers, however, will find that asymptotic theory is short-changed.
The author states the weak law of large numbers but gives no proof, the strong law of large numbers is not mentioned, and the discussion of the central limit theorem will not be particularly enlightening for most undergraduate probability students. The author concludes the book with chapters on unbiased estimation, sampling, and generating random numbers. Designed to show the reader how probability theory applies to statistics, these chapters are short and present only a brief overview of the areas. They might be difficult to use directly in a standard course in probability theory, but they would be good extra reading for an interested student. The publication format of the book is less desirable than that of many modern introductory textbooks.
The print is small; there is no use of color and very limited use of graphics. The format does little to motivate a mildly interested student to read the book. SPURRIER University of South Carolina Fundamentals of Queueing Theory (2nd ed.). Donald Gross and Carl M. New York: John Wiley, 1985. Xiii + 587 pp.
This second edition of the authors' 1974 book reviewed by Moore (1977) contains some new material and expanded versions of old ma- terial. Queuing models are introduced in Chapter 1, which also contains a review of the Poisson process and new material on fundamentals of Markov processes.
Chapter 2 incorporates material from the old Chapters 2 and 3, treating single and multiserver queues with Poisson arrivals and exponential service times; in addition, the basic theory of Markov chains in discrete and continuous time is outlined. Material from the old Chapter 4 is now expanded into Chapters 3 and 4. Bulk queues, Erlangian systems, and priority queues are treated in Chapter 3, which also contains a brief mention of phase-type distributions. Chapter 4 treats networks, including queues in series and cyclic queues. Embedded Markov chains are treated in Chapter 5, with a brief introduction to the matrix-geometric concept. Old Chapters 6 and 7 are now considerably expanded and reorganized. Chapter 6 treats the general single-server queue and miscellaneous topics such as semi-Markovian queues, design and control of queues, and sta- tistical inference from queues.
Chapter 7 deals with bounds, approxi- mations, and numerical solutions. The final Chapter 8 contains an ex- panded treatment of simulation techniques for queues. Missing from the second edition is the old Chapter 9 on a case study; a more recent case study would have been useful, or the old material should have been retained.
The book is written in an easy, flowing style. Technical parts of the proofs are explained in great detail (sometimes in too much detail). In every chapter there are illustrative examples and a fairly large set of problems for solution. A touch of humor is present in the description of examples and problems, but no flippancy is intended. A background of calculus, stochastic processes, and statistics is assumed (if persons with this background wish to call themselves nonmathematicians, that is fine, but queuing theory is essentially a mathematical theory). One might complain about one's favorite topics being left out of this book, but in my opinion the authors' treatment of basic queuing theory is perhaps the most satisfactory among books at this level.
This book will serve as a useful book for an introductory course on the subject and a valuable reference to practitioners. PRABHU Cornell University REFERENCE Moore, Roger H. (1977), Review of Fundamentals of Queueing Theory, by Donald Gross and Carl M. Harris, Journal of the American Statistical Association, 72, 232-233. This content downloaded from 62.122.76.48 on Sun, 15 Jun 2014 15:48:04 PM All use subject to JSTOR Terms and Conditions Article Contents p. 947 Issue Table of Contents Journal of the American Statistical Association, Vol. 399 (Sep., 1987), pp.
I-iv+705-963 Front Matter pp. Applications An Application of Bayes Methodology to the Analysis of Diary Records from a Water Use Study pp. 705-711 Restricted Randomization: A Practical Example pp. 712-719 On Concurrent Seasonal Adjustment pp.
720-732 The Effects of Annual Accounting Data on Stock Returns and Trading Activity: A Causal Model Study pp. 733-738 Theory and Methods Empirical Bayes Confidence Intervals Based on Bootstrap Samples pp. 739-750 Empirical Bayes Confidence Intervals Based on Bootstrap Samples: Comment pp. 751-752 Empirical Bayes Confidence Intervals Based on Bootstrap Samples: Comment pp.
752-754 Empirical Bayes Confidence Intervals Based on Bootstrap Samples: Comment pp. 754 Empirical Bayes Confidence Intervals Based on Bootstrap Samples: Comment pp. 755-756 Empirical Bayes Confidence Intervals Based on Bootstrap Samples: Rejoinder pp. 756-757 A Unified Treatment of Integer Parameter Models pp. 758-764 Minimum Norm Quadratic Estimation of Spatial Variograms pp. 765-772 Bayesian Methods for Censored Categorical Data pp.
773-781 An Exact Test for Multiple Inequality and Equality Constraints in the Linear Regression Model pp. 782-793 Time- and Space-Efficient Algorithms for Least Median of Squares Regression pp. 794-801 Minimum Hellinger Distance Estimation for the Analysis of Count Data pp. 802-807 Regression Methods for Poisson Process Data pp. 808-815 Test Statistics Derived as Components of Pearson's Phi-Squared Distance Measure pp. 816-825 Conditioning Ratio Estimates Under Simple Random Sampling pp.
826-831 A Frequency-Domain Median Time Series pp. 832-835 Identifying a Simplifying Structure in Time Series pp. 836-843 Marginal Curvatures and Their Usefulness in the Analysis of Nonlinear Regression Models pp. 844-850 L-Estimation for Linear Models pp. 851-857 A Semiparametric Approach to Density Estimation pp.
858-865 A Comparison of Variance Component Estimates for Arbitrary Underlying Distributions pp. 866-874 Quick Simultaneous Confidence Intervals for Multinomial Proportions pp. 875-878 Model Robustness for Simultaneous Confidence Bands pp. 879-885 Best Median-Unbiased Estimation in Linear Regression with Bounded Asymmetric Loss Functions pp. 886-893 Minimax Regret Simultaneous Confidence Bands for Multiple Regression Functions pp. 894-901 Comparison of Several Treatments with a Control Using Multiple Contrasts pp.
902-910 Influence Analysis of Generalized Least Squares Estimators pp. 911-917 K-Sample Anderson-Darling Tests pp.
918-924 How Much Better Are Better Estimators of a Normal Variance pp. 925-928 Small Sample Properties of Probit Model Estimators pp. 929-937 Parameter Estimation for the Sichel Distribution and Its Multivariate Extension pp. 938-944 Book Reviews List of Book Reviews pp. 945 Review: untitled pp. 946 Review: untitled pp. 946-947 Review: untitled pp.
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962-963 Back Matter pp.