**The Theory of Statistical Inference (1937), by Samuel Stanley Wilks.****1.1. Review by: Maurice George Kendall.**

*Journal of the Royal Statistical Society***101**(2) (1938), 457-458.To the historian of statistics a hundred years hence the present time will probably appear as an Age of Discovery. We live at a period of intense development, when new statistical territory is being opened up daily, vast areas of virgin soil are made available for the eager pioneer, and the general field of research is a sort of Tom Tiddler's ground where even the amateur may dig with profit. Against the gratification with which one is tempted to contemplate these happy conditions there must, however, be put the difficulty experienced by the average statistician in keeping abreast of new discoveries and fitting them into his scheme of things. Many of the important contributions to statistical theory of the past thirty years are still hidden away in scientific periodicals. The knowledge is there, but it has hardly been disseminated beyond the lecture-room. ... In the meantime such notes as these of Dr Wilks are very welcome. They relate to a set of lectures to graduate students at Princeton, and are an effort "to develop systematically, together with illustrative examples, some of the essential ideas involved in the theory of statistical inference." ... Dr Wilks allows himself a full mathematical technique, and he is, in fact, concerned rather with the mathematical development of the consequences of the ideas than with their logical justification. Nevertheless this work will be of considerable service to those wishing to embark on the study of advanced statistical theory

**Mathematical Statistics (1943), by Samuel Stanley Wilks.****2.1. Review by: Tjalling Koopmans.**

*Journal of Political Economy***52**(3) (1944), 277-279.Mathematical statistics - the art of drawing inferences from statistical observations - has come to be developed and used in relation to many different domains of application. Based on mathematics and on logic, its underlying principles and procedures are the same in each field to which it is applied. But its literature is partly scattered among many journals devoted principally to other sciences and partly concentrated in a few specialized periodicals and memoirs read more by those engaged in developing this young science than by those who are interested in it as an aid to their chosen field of research. This state of affairs has been inevitable, in view of the rapid growth of statistical theory. ... Professor Wilks has done a great service to the cause of statistical theory by making available in book form notes from a two-semester course given at Princeton University. This book is a first step toward a systematic exposition of the mathematical theory of statistics and will provide a most valuable framework for further systematizing efforts. The text is written in the crisp language of mathematical reasoning and will be enjoyed most by those who have trained themselves in reading that form of prose. ... Readers of this Journal will desire to know in what respects the book and its subject matter are useful to economists. ... The general answer to the question ... is that this book contains: (a) a succinct and dear exposition of the general theory of statistical inference, and (b) a wealth of specific theory designed to meet the conditions of a great many fields of application. In the contents of this book economists thus possess both a general theoretical framework and instructive examples, which will be of invaluable assistance when they set out to develop the statistical theory specifically adapted to the estimation problems concerning economic time series and the relations between them.

**2.2. Review by: H L S.**

*Journal of the Institute of Actuaries*(1886-1994)**72**(2) (1946), 295-296.Prior to Wilks's book under review, no text-book on mathematical statistics had given prominence to the theories of estimation and tests of statistical hypo- theses which are so fundamental to the important present-day applications of mathematical probability. These theories date from Fisher's classical papers in the early 1920's and the Neyman-Pearson work published about 1930, and require for their understanding many of the previously developed concepts relating to distribution functions. However, the basic mathematics is now of degree standard and it has yet to be seen whether these theories can be satisfactorily understood by a student without considerable mathematical knowledge. Wilks's book, in fact, is based on a series of lectures delivered at Princeton to graduate and advanced undergraduate students of mathematics. These lectures seem to have been the lineal descendants of those on which his previous monograph, Statistical Inference (1937), was founded, though there are some interesting differences of outlook noticeable in the two treatments where they over- lap. ... To anyone who wishes to know what problems the statistician of to-day is successfully tackling, Wilks's excellent book will be indispensable. It provides the only comprehensive, step-by-step, mathematical treatment of modern statistical technique.

**2.3. Review by: Abraham Wald.**

*J. Amer. Statist. Association***38**(224) (2943), 491-492.The study of Mathematical Statistics has been seriously hampered by the lack of good books and by the fact that many important developments of the theory can be found only in the original form and widely scattered in scientific periodicals. The author has performed a very valuable service to all interested in the field by writing the present excellent book. The content of the lithoprinted book is formed chiefly by the subject matter of a two-semester course given by the author for advanced undergraduate and beginning graduate students. As such it has, of course, its limitations. It was impossible to cover in great detail all important developments of the theory or to give very extensive illustrations. However, the book contains the basic material in a concise form and gives an excellent introduction to the more recent developments of the mathematical theory of statistics. ... Readers with some mathematical background will find the book an excellent introduction to the modern developments in the theory of mathematical statistics. Also teachers of mathematical statistics will find it very helpful in their classroom work. The book can be highly recommended to everyone interested in this field.

**2.4. Review by: Maurice Stevenson Bartlett.**

*Journal of the Royal Statistical Society*106 (3) (1943), 279-280.In 1937 the author, by the publication of his course of lectures at Princeton University on the mathematical theory of statistical inference, was perhaps the first to produce an authoritative work on this subject. This new lithoprinted publication, which contains nearly three times as many pages as its predecessor, is similarly based on a lecture course. The author, by suggesting in the preface that "the present notes" are still to be "revised and issued in permanent form," and by noting further that some omissions are dictated by the particular needs of his students, rather tends to disarm possible criticism on subject-matter and layout. However, from the title the general reader might be entitled to expect a comprehensive treatment. It is for him that this review has been written. ... This book will be useful to the professional mathematical statistician; also to mathematically-minded students provided that they are not solely reliant on it; it is not recommended to the amateur, nor to the student who is more interested in method than in theory.

**2.5. Review by: Eugene Grebenik.**

*Econometrica, New series***14**(55) (1947), 239-241.Dr Wilks's book, which appeared some time ago, is severely theoretical in its nature. It covers most of modern statistical theory, but no examples are given, and the reader is expected to have a good mathematical equipment. The chapters on the theory of estimation and on multivariate analysis are particularly valuable.

**2.6. Review by: Jerzy Neyman.**

*Mathematical Reviews*, MR0008657**(5,41d)**.In a sense the book under review is unique. The great number of books on the market dealing with statistics and probability is notorious. All these fall roughly into two categories. The majority describe "statistical methods,'' frequently ignoring the basic ideas and the mathematics behind these methods, occasionally misinterpreting them. The books of the other category, some of them excellent [Borel Series, Cramér, Kolmogorov, Uspensky], deal with mathematical theory of probability with only occasional glimpses on some particular questions pertaining to statistical theory. In contrast, the aim of Wilks is to give the students an organized source of information concerning the theory of statistics as such. The need of such a source is great and the book under review deserves general recognition. The book, covering the author's courses at Princeton University, was compiled with the collaboration of H Scheffé, T W Anderson and D F Votaw. ... The book as a whole is a very useful one. As mentioned, it is as yet the only book where a student can get information on the theories of testing hypotheses and on estimation. Besides, it contains a wealth of information on the distributions of various important statistics. ... The book would probably gain if it were supplemented by a few more examples, particularly with a few numerical examples. However, as it stands, it is an excellent addition to modern statistical literature.

**Elementary Statistical Analysis (1948), by Samuel Stanley Wilks.****3.1. Review by: Alphonse Chapanis.**

*The Quarterly Review of Biology***24**(2) (1949), 180.A large number of statistics textbooks seems to have appeared in the last few years. For the most part, they all cover roughly the same content but differ in approach. This one, designed for a one-semester course, differs from most in that it is characterized by heavy emphasis upon sampling theory and techniques. Long before the student has read a quarter of the way through the book, he is deep in a discussion of probability. This is a stiffer dose of probability and sampling than one finds in most elementary books. Wilks' approach has much to commend it, and a choice among textbooks depends ultimately on what individual instructors consider important. This one, as might be expected, is light on the descriptive statistics. ... Although the author claims that the text "presupposes one semester of elementary mathematical analysis," the book requires a better mathematical background than many others at a com-parable level.

**3.2. Review by: Frank Sandon.**

*The Mathematical Gazette***33**(304) (1949), 154.S S Wilks was the author of Mathematical Statistics: the present volume refers to this earlier one as a "more advanced book on this subject". The new book is prepared for a one-semester basic course (usually taken at Princeton in the freshman year). ... The book can be highly recommended. It is sound and scholarly. The questions are good, though there are no answers. The typescript itself seems to me to be much clearer than was that of Mathematical Statistics: perhaps this is because the earlier book included many heavy mathematical formulae. There are few misprints. There is a useful index but no bibliography. The book is based on some of Wilks' original work and is an original treatment of the subject: but though it is described as "elementary" it will put the reader on the right lines.

**3.3. Review by: Theodore Alfonso Bancroft.**

*J. Amer. Statist. Association***44**(247) (1949), 458-459.Professor Wilks has attempted with the material and arrangement of this new text to follow the recommendations of three distinguished committees on the teaching of statistics regarding the introduction of a basic elementary course available centrally to all students needing an understanding of statistical concepts and techniques common to all fields of application. Quoting from the preface, Professor Wilks says, 'This book has been prepared for a one-semester basic course in elementary statistical analysis which, at Princeton, is the introductory course for all fields of statistical application, and is usually taken in the freshman year. It is especially designed for those who intend to go into the biological and social sciences. It presupposes one semester of elementary mathematical analysis covering topics such as those included in the first half of F L Griffin's "Introduction to Mathematical Analysis." The text, then, is designed as a basic general service course, presumably to be offered in a mathematics or statistics department at the latter part of the freshman year. ... Professor Wilks' experiment in teaching such materials in a central freshman-level course should be observed closely. It represents an approach by a mathematical statistician to provide teaching material for this type course. Attempts to satisfy this need from another approach are being made at other institutions by applied or experimental statisticians. In the reviewer's opinion, the two approaches should supplement rather than compete with one another. Possibly ideal teaching materials for such a course will be the outcome of cooperative efforts of several statisticians with as many viewpoints.

**3.4. Review by: Erich Leo Lehmann.**

*Amer. Math. Monthly***56**(6) (1949), 429-430.This book is intended for a one semester elementary course in statistics. As the author states in the preface, he has been teaching this material as an introduction to statistics to students in all fields of application. In accordance with such a purpose only a very modest mathematical background is presupposed, essentially just the concepts of derivative and integral, such as a student might acquire in a one term calculus course. It is clear that on this basis a complete treatment of much of the theory of statistical inference is not possible. The common way out of this difficulty is to make such a text a collection of methods, stated as a set of directions, together. ... with illustrations of their use. The present book-breaking away from this unfortunate tradition-has as its goal the understanding of statistics rather than manipulative proficiency. As the foundation for such understanding the elementary (discrete) theory of probability is developed. This includes in particular the more important theorems concerned with the notions of addition, multiplication and complementation of events, and of expectation and variance of a random variable. This material, together with a discussion of the binomial and Poisson distributions and with some extensions to the continuous case, forms the central part of the book, taking up about one half of the space. ... By providing an elementary course that can be taught to students in many fields of application, and in which the objective is the teaching of concepts rather than rules, Professor Wilks has made an important contribution to elementary instruction in statistics. One may also hope that his book will en- courage the centralization, within universities, of elementary statistical teaching.

**3.5. Review by: Thomas Nall Eden Greville.**

*Science, New Series***109**(2835) (1949), 450.This book is intended to be used as a text in a one-semester introductory course in statistics. While a knowledge of calculus is assumed, it is used only rarely: an acquaintance with elementary algebra will suffice for the reading of all but a few sections of the book, and those could be omitted without serious loss. It is clearly and lucidly written, and numerous examples are provided in the text to illustrate the principles brought out. The aim has been to make a few basic concepts entirely clear, rather than to cover a wide field. The viewpoint is adopted which regards statistical analysis as a methodological tool of scientific research, rather than the traditional idea, still too common in elementary texts, that its main object is merely to give a summary description of a set of data. Thus, the role of probability is emphasized; and much attention is given to the problem of sampling, which is that of making inferences from a sample concerning the characteristics of the population from which it was drawn. It is refreshing to find that, by the introduction of confidence limits, the author has brought into his treatment of this subject a definiteness too often lacking in beginning textbooks.

**Mathematical Statistics (1962), by Samuel Stanley Wilks.****4.1. Review by: Roger S Pinkham.**

*Amer. Math. Monthly***69**(9) (1962), 937.This book is structured much like the book of the same name by the same author which was first published in 1953. The new book contains vastly more information and is more tightly knit. Although the result of a powerful compression, I believe it will prove to be readable by students. ... The powerful compression and meticulous organization of the book constitute both a strength and a weakness. The inexperienced student may think that nothing remains to be done. Then too, since the author has purposely omitted discussion of a data analysis and methodology, it is possible to get a biased view of the field as a whole. ... This is a remarkable book which is destined to be a classic in the field. The author is to be praised for such an immense effort so carefully carried out.

**4.2. Review by: Peter Whittle.**

*Journal of the Royal Statistical Society. Series A (General)***126**(1) (1963), 128-129.Professor Wilks's book offers a course, at once comprehensive and concentrated, in mathematical statistics at a medium advanced level, together with the requisite probability theory. This is sufficient to characterize it as unique, for there is no other single-volume work for which the same claim can be made. ... It will be clear that the book is both comprehensive and systematic. The treatment is consistently close and technical, with the result that the 644 pages contain an impressive amount of detailed material, complete with proofs. On this last point, the author seems to have resolved to give a uniform treatment, with no points shirked, so that, for example, most of the limiting and asymptotic results are proved completely and under explicit conditions. ... A rather surprising feature is that only slight use is made of matrix concepts, and scarcely any of matrix notation. The use of the notation alone would have improved the presentation and greatly lightened the typography. But, of course, the book stands (and requires) criticism only because it is so unquestionably substantial and important. Its virtues bear reiteration: it is completely without padding or superficiality, and so is unique in the extent and detail of its coverage; it consistently maintains a good level of proof, often by the use of original arguments; it is for the most part easy to read, and accessible as a reference. It is the sort of book that one uses a great deal once one has it, and, even on a student's scale of values, is well worth the rather daunting purchase price.

**4.3. Review by: D E Barton.**

*Biometrics***19**(1) (1963) 194-195.This is a very valuable book. Born of the book of the same name written in 1943 and which has been a standby of statisticians lucky enough to have a copy ever since, it is only just recognizable as the son of its father. This is not only due to the vast growth of statistics over the last twenty years and the fact that the book has taken ten years to write, but also to its wider and more ambitious conception. What remains is the author's excellent judgement as to what is and what is not important in statistics. It is a book for the mathematics graduate with some familiarity with the Lebesgue-Stieltjes integral and related measure-theoretic notions and, to this extent, is not self-contained. ... Most of the content (but not its treatment) of the earlier chapters would have to be given in any book of this title and one can only say how judiciously the ingredients have been mixed and how well served. As one would expect from the author of Wilks's theorem, the chapter on asymptotic theory of maximum likelihood, likelihood-ratio and allied statistics is particularly clear and thorough.

**4.4. Review by: R Coppi.**

*Genus***22**(1/4) (1966), 399-400.This book, taken as a whole, constitutes a rigorous and essential presentation of mathematical foundations of statistical methodology based on the use of random samples. It is the result of an in-depth study of layout which has lasted for almost twenty years since the first edition, which appeared in the form very small compared to the current one, of 1943.

**4.5. Review by: Gustav Elfving.**

*Mathematical Reviews*, MR00144404**(26 3 1949)**.This important book is, in the author's own words, intended to introduce mathematical statistics to readers with good undergraduate backgrounds in mathematics. In the reviewer's opinion, it is hardly an introductory book, but certainly an excellent text-book on an advanced level, and most likely to become a valuable source of reference for research workers in theoretical as well as applied statistics. ... As the author states in his preface, he has tried to make a unified and systematic presentation of classical results of mathematical statistics, without going into too many ramifications. This he has accomplished with great skill. ... The concise style, the carefully chosen notation, and the emphasis on mathematical methods, as opposed to statistical techniques, remind the reader of Cramér's classical book [

*Mathematical methods of statistics*]. The author has more material and somewhat less philosophy. The details are sometimes complex; the totality is of appealing simplicity and homogeneity.**Introductory Engineering Statistics (1966), by Irwin Guttman and Samuel Stanley Wilks.****5.1. Review by: W G Gilchrist.**

*Journal of the Royal Statistical Society. Series A (General)***129**(4) (1966), 593-594.The stated object of this book is "to give undergraduate engineers a facility in, and understanding of, some elementary applied statistical techniques". After initial chapters on probability and discrete distributions a chapter is devoted to the applications of these in acceptance sampling. Further chapters on the description of samples, probability distributions and the normal distribution lead again to specific applications in acceptance sampling. The next two chapters give introductions to estimation and statistical testing in their "classical" form. Then follow chapters on control charts, chi-square goodness-of-fit tests, order statistics and non-parametric tests. The book ends with a standard treatment of regression analysis and the analysis of variance. ... One very good aspect of the treatment of the book is that though each topic is treated somewhat mathematically, each result is immediately followed by a worked numerical example. If a student conscientiously works through the problems and has them corrected, he should certainly achieve the "facility" the authors aimed at. One suspects, though, that engineers would require supplementary help to reach the "understanding" aimed at. However, for the reader used to the mathematical presentation of ideas the exposition is on the whole clear and concise. The book is thus a useful addition to the fairly small group of texts written for engineering students.

**5.2. Review by: Michael F Capobianco.**

*Technometrics***8**(3) (1966), 554-555.The purpose of this book as stated in the preface is to give undergraduate engineers a facility in, and understanding of, some elementary applied statistical techniques. It is also stated that no previous background in probability and statistics is assumed, but that at least one introductory course in calculus is required. Used as a text by a competent instructor, the book should succeed quite well in accomplishing its purpose. However, it is generally very concise, and in some places poorly motivated, so that it is doubtful that an engineer having no knowledge of probability or statistics would be able to read it successfully on his own. Should he be able to do so, the rewards for his efforts would be substantial, for the book covers a good many techniques useful to engineers, and in sufficient theoretical depth (especially if one reads appendices I-VI) to provide a firm understanding. ... In summary, we have in this book a fairly good text for a first course designed to give engineers a combination of the theory and practice of statistics. We do not have a very good book for self study.

**Introductory Engineering Statistics (2nd Edition) (1971), by Irwin Guttman, Samuel Stanley Wilks and J Stuart Hunter.****6.1. Review by: M Stone.**

*Journal of the Royal Statistical Society. Series A (General)***136**(2) (1973), 262.This book is designed to meet a need which is not manifest on this side of the Atlantic - the teaching of statistics to large numbers of engineering students. (Do our light-bulbs, computers and bridges have a carte-blanche to keep them untouched by the hand of probability?) It is therefore somewhat presumptuous of me to write that the book appears to fulfill admirably its stated objectives. Its style clearly owes much to its association with the late Professor Wilks; it does not beat about the philosophical bush but, quite appropriately gets down to brass tacks. The treatment is standard and covers topics outside stochastic processes that one would expect to find. This second edition has the novelty of a chapter on Bayesian estimation. There is also a ell-balanced new chapter on experimental design.

**6.2. Review by: A Grandage.**

*Technometrics***14**(4) (1972), 979.This book is an expanded version of the first edition which was reviewed in Technometrics for August 1966. Two hundred pages of new material have been added including two new chapters-one on Bayesian Estimation and one on Analysis of 2

^{k}factorial designs and response surfaces. ... With reservations, this text is recommended for a two-semester sequence to give undergraduate engineers a facility in, and understanding of, some elementary applied statistical techniques. It assumes one course in calculus but no previous training in probability or statistics. The main criticisms of the first edition, lack of motivation and a concise style with concentration on the mathematics of the subject have not been overcome in this expanded edition with the exception of the new chapter on the analysis of factorial and response surface designs. In the hands of an instructor with experience in applied statistics to motivate and indicate engineering application, this book could be the basis of a very rewarding course. Without an instructor or previous experience in statistics, the text is not recommended.