PSYCHOMETRIKA~VOL. 15, NO. 4 DEC~BER, 1950

BOOK REVIEW (eft.) P~'ooe6dings of the Berkeley Symposium on Mathematical Statistics and Probability. Berkeley: University of California Press,

JERZY NEYMAN

1949. Pp. 501 + viii.* The Berkeley Symposium on Mathematical Statistics and Probability, made possible by a special g r a n t of the Administration of the University of California, brought together, during August, 1945, and J a n u a r y , 1946, a total of thirty-three specialists and tapped their knowledge of mathematical statistics and probability, theory and applications. These Proceedings are the tangible product of this effort, and consist of two p a p e r s on the logical and philosophical foundations of probability and statistical inference; seven on analytical and computational mathematics of probability and statistics; six on the theory and techniques of statistical inference as such; one on the teaching of statistics and its role in a university, with supplementary discussion; and thirteen others on applications of probability and statistics. Some of the papers, both theoretical and applied, present brand-new results; some, neater solutions to previously solved problems; and some review p a r t i c u l a r phases of probability and statistics. Applications considered explicitly, or alluded to, relate to problems in agriculture, animal breeding, astronomy, demography, economics, entomology, evolution, forestry, genetics, insurance, metcorology, m i l i t a r y strategy, philosophy, physics, population dynamics, psychology, and telephony. Discussion of some of these is limited to presentation of experimental and operational problems requiring statistical t r e a t m e n t ; others serve to illustrate the use of probability and statistical theory as tools in the planning, conducti and interpretation of experiments and inquiries relating to complex phenomena. All in all, this volume provides an interesting, valuable, and stimulating panoramic view of nrobabitity, mathematical statistics, and their applications as of J a n u a r y , 1946. I t is a " m u s t " for libraries of institutions where research is conducted a t the postgraduate level in the fields covered by the p a p e r s included. The individual scientist will probably find only a few papers of direct interest to him and will be content to rely on a l i b r a r y copy, unless he is intimately concerned with mathematical statisties~ A few of the papers of more general interest a r e discussed briefly below. "The Place of Statistics in the University," ~y Professor Harold Hotelling, is his second p a p e r on the teaching of statistics and its role in a university o r college. (The first, "The Teaching of Statistics," has been published in the Anneals of Meth~r~ticaZ Statistics, Vol. II, No. 4, December, 1940). In summing up his paper, Professor Hotelling states: The teaching of statistics, which has grown r a p i d l y and seems likely to grow much f u r t h e r still, has many unsatisfactory features. The chief of these is the inadequate preparation in statistical theory of a large proportion of those teaching the subject. The evils tend to be perpetuated by the prevailing system of independent courses in elementary statistical method scattered through numerous departments concerned with applications. This system places the selection, supervision, and promotion of teachers of statistical method and theory in the hands of those who a r e not specialists in this subject. Teachers and prospective teach*The R e v i e w E d i t o r a c k n o w l e d g e s w i t h r e g r e t t h a t t h i s r e v i e w is a c o n d e n s a t i o n , m a d e

neee s s a r y b y s p a c e limitations, of a m o r e l e n g t h y r e v i e w w h i c h D r . E l s e n h a r t h M w r i t t e n . T h e c o m plete r e v i e w c o n t a i n s a m o r e detailed dlseusslon o f m a n y o f t h e articles, f r e q u e n t l y w i t h e x t e n s i o n s b y t h e r e v i e w e r , a n d h i s t o r i c a l orlentation,~. Copies o f t h e c o m p l e t e r e v i e w a r e a v a i l a b l e b y d i r e c t correspondence w i t h D r . E i s e n h a r t ,

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PSYCHOMETRIKA ers of the theory of statistics feel a pressure to divert their efforts away from this theory and into its applications. In consequence, both statistical theory and the underlying mathematics are slighted, with the result t h a t erroneous and inefficient methods continue to be t a u g h t and applied. It is recommended that the preparation of teachers of statistical methods and theory be focused more definitely on t h i s subject itself and the mathematics essential to it. Some study of a field of application, and practice in applications, are also desirable, but should not dominate the graduate curriculum in statistics. Organization of the teaching of statistical methods should be contralized, and should provide also for the joint functions of research and of advice and service needed by others in the institution, and possibly outside it, r e g a r d i n g the statistical aspects of their problems of designing experiments and interpreting observations. Beginning courses in statistical methods and theory should be t a u g h t only under the supervision of the central statisl;ical organization, but courses in applied statistics, requiring these beginning courses as prerequisites, might be taught in any department. Of these first courses there should be two, one based on calculus and the other requiring no mathematics beyond elementary algebra. The more mathematical of these courses would be the more valuable, and efforts should be made to bring the l a r g e r number of students into it. The central statistical group would also teach more advanced courses in the subject.

Each time I have read Professor Hotelling~s papers on the teaching of statistics I have come away with two strong impressions: first, t h a t he is primarily interested in the development of statistics as a branch of mathematics and the production of statistical scholars: and second, t h a t he ignores the problem of t r a n s f e r of ideas, habits, and skills from one context to another. His insistence on a central first course in statistical m e t h o d - - o r two such central courses, one requiring calculus and the other no mathematics beyond elementary a l g e b r a - is justified at present, I feel, by the acute shortage of persons adequately trained in modern statistical methods within the various subject-matter fields where statistical method can be used to great advantage. As "statistical missionaries," trained in such "first courses" and their sequels, become available in those fields, students should receive their" first courses from these "missionaries," who will be better fitted to reveal statistical method as a means to ends l vin~r within the subject-matter field by considering problems of intrinsic interest to the class, by utilizing real data of familiar types, by assigning more work on material in the r a w and less tidying-up of textbook exercises, and by pointing out simple approximate methods that lead to the same conclusions or doubts as more elaborate methods involving, perhaps, more restrictive assumptions. When this day has come, the more mathematical of Professor Hotelling's "first courses" might then be retained as the sole first course in statistics for prospective mathematics and statistics majors, and constitute a "second course" for students of other fields who wish to uursue statistics beyond their first courses. The ultimate need f o r "first courses" spread around the campus in addition to a central more mathematical "first course" a r r e a r s to he attested to by several of Professor Hotening's discussants, who express a clear recognition of the fact t h a t students who have had little or no research experience more readily acquire an appreciation - - and a better understanding! m of statistical methods when the problems discussed and the illustrative data are drawn from familiar fields than when statistics is taught as a separate discipline. One of the most important ,papers in the Proceed~os. from the viewpoint of techniques of statistical inference, is Professor Wolfowitz's contribution, "N'on-parametric Statistical Inference." By way of introduction he writes, " I n most statistical problems treated in the literature a datum of the problem is the information that the various distributions of the chance variables involved belong to given families of distribution functions (d.f.'s) completely specified except f o r one or more parameters. Non-parametric statistical inference i s coneerned With problems where the d.f.'s are not specified to such an e x t e n t , and

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w h e r e t h e i r f u n c t i o n a l f o r m is u n k n o w n . T h i s does n o t p r e c l u d e some k n o w l e d g e of t h e d.f.'s; f o r e x a m p l e , we m a y k n o w t h a t t h e y a r e c o n t i n u o u s , uni-modal, bimodal, a n d t h e like." P r o f e s s o r W o l f o w i t z ' s p a p e r p r o v i d e s a n i n t e r e s t i n g a n d r e a d a b l e s u r v e y of n o n - p a r a m e t r i c p r o c e d u r e s , w i t h a t t e n t i o n to s u c h m a t t e r s a s t h e " c o n s i s t e n c y " a n d "efficiency" of n o n - p a r a m e t r i c p r o c e d u r e s . H i s is a " m o r e or less h e u r i s t i c a n d i n t u i t i v e " p r e s e n t a t i o n , m a k i n g " n o a t t e m p t a t c o v e r i n g t h e e n t i r e field." I n a n a p p e n d i x to his p a p e r , P r o f e s s o r W o l f o w i t z c o n t r i b u t e s a n e w r e s u l t : a f o r m u l a f o r t h e a s y m p t o t i c v a r i a n c e of U, t h e t o t a l n u m b e r of r u n s i n a s e t of o b s e r v a t i o n s , w h e n t h e o b s e r v a t i o n s a r e d r a w n f r o m two d i f f e r e n t p o p u l a tions. T h i s r e s u l t m a k e s i t possible to o b t a i n a good a p p r o x i m a t i o n to t h e p o w e r f u n c t i o n of the W a l d - W o l f o w i t z t e s t of w h e t h e r two s a m p l e s come f r o m t h e s a m e p o p u l a t i o n " f o r a l t e r n a t i v e s s u b j e c t to some s l i g h t . . . r e s t r i c t i o n s , w h e n t h e s a m p l e sizes a r e l a r g e . " Dr. P. L. H s u ' s c o n t r i b u t i o n , " T h e L i m i t i n g D i s t r i b u t i o n of F u n c t i o n s of S a m p l e M e a n s a n d A p p l i c a t i o n to T e s t i n g H y p o t h e s e s , " is a v e r y v a l u a b l e a d d i t i o n to t h e l i t e r a t u r e on a s y m p t o t i c d i s t r i b u t i o n s of f u n c t i o n s of s a m p l e mom e n t s - - a l i t e r a t u r e c o n t a i n i n g a w e a l t h o f r e l a t e d r e s u l t s w h i c h someone s h o u l d t a k e t h e t r o u b l e to consolidate f o r t h e benefit of all of us. I n a p r e v i o u s p a p e r , * Dr. H s u g a v e some t h e o r e m s f o r t h e case of f u n c t i o n s of " m e a n s " ( i n t e r p r e t e d i n - t h e - l a r g e , so as to include m o m e n t s a n d p r o d u c t m o m e n t s ) of a n y finite n u m b e r of s a m p l e s of t h e s a m e size. I n t h e first p a r t of his Proceedings p a p e r , h e gives two t h e o r e m s w h i c h e x t e n d t h e s e r e s u l t s to t h e case of " m e a n s " of s a m p l e s of d i f f e r e n t sizes, h i s discussion b e i n g l i m i t e d h e r e ( a n d i n h i s p r e v i o u s p a p e r ) to s i t u a t i o n s w h e r e t h e l i m i t i n g d i s t r i b u t i o n is n o r m a l or t h e d i s t r i b u t i o n q u a d r a t i c f o r m in n o r m a l v a r i a b l e s , n o t i n g , in t h e l a t t e r i n s t a n c e , t h e c i r c u m s t a n c e s u n d e r w h i c h t h e d i s t r i b u t i o n reduces to a x ~ d i s t r i b u t i o n . E x a m p l e s c o n s i d e r e d explicitly include t h e x 2 t e s t f o r goodness of fit, t h e x 2 t e s t of " h o m o g e n e i t y " in c o n t i n g e n c y tables, S t u d e n t ' s t, t h e L a n d L 2 t e s t f u n c t i o n s , a n d W i l k s ' t e s t f u n c t i o n s f o r t e s t i n g t h e i n d e p e n d e n c e of k sets of r a n d o m v a r i a b l e s . I n t h e second p a r t of h i s p a p e r , Dr. H s u c o n t i n u e s , in m u c h t h e same vein, w i t h a c o n s i d e r a t i o n of H o t e l l i n g ' s T a n d M a h a l a n o b i s ' D (in b o t h s t u d e n t i z e d a n d u n s t u d e n t i z e d f o r m s ) a n d f o r m u l a t e s a s y s t e m a t i c m e t h o d of c o n s t r u c t i n g t e s t f u n c t i o n s f o r e i t h e r of two h y p o t h e s e s o f g e n e r a l c h a r a c t e r r e l a t i n g to m u l t i p l e s a m p l e s f r o m m u l t i v a r i a t e d i s t r i b u t i o n s . T h e t i t l e of P r o f e s s o r N e y m a n ' s own p a p e r , " C o n t r i b u t i o n to t h e T h e o r y of t h e x 2 T e s t , " does n o t r e v e a l w h a t t h e p a p e r is r e a l l y about. W h i l e s e v e r a l a l t e r n a t i v e definitions of t h e f a m i l i a r symbol x 2 a r e discussed, t h e body of t h e p a p e r is divided into two p a r t s , t h e first defining a n d d i s c u s s i n g " a class of e s t i m a t e s • . . t e r m e d best asymptotically normal estimates ( B A N e s t i m a t e s , f o r s h o r t ) , all h a v i n g t h e s a m e a s y m p t o t i c p r o p e r t i e s as t h e m a x i m u m - l i k e l i h o o d e s t i m a t e s b u t v a r y i n g in t h e ease w i t h w h i c h t h e y c a n be c o m p u t e d " [ i t a l i c s o u r s ] , a n d t h e second devoted to d e v e l o p m e n t of " a class of t e s t s . w h i c h a r e all equival e n t in t h e l i m i t to ~ t e s t s . " A l t h o u g h t h e p a p e r is p r i m a r i l y concerned w i t h t e c h n i q u e s of e s t i m a t i o n a n d t e s t i n g s t a t i s t i c a l h y p o t h e s e s t h a t a r e o p t i m a l - w i t h p e e r s b u t no s u p e r i o r s , a n d h e n c e equally e n t i t l e d to be called " b e s t " w h e n b a s e d on infinitely la~'ge samples, t h e t i t l e a d o p t e d is n e v e r t h e l e s s a p p r o p r i a t e since " b o t h t h e c o m p u t a t i o n of B A N e s t i m a t e s a n d t h e a p p l i c a t i o n of t h e s t a t i s t i c a l t e s t s c o n s i d e r e d involve m i n i m i z a t i o n of t h e a l t e r n a t i v e l y defined x2's. '' A w a r n i n g is sounded b y P r o f e s s o r L e h m a n n in h i s p a p e r , " S o m e C o m m e n t s on L a r g e S a m p l e T e s t s , " a g a i n s t c o n c e n t r a t i n g too m u c h a t t e n t i o n on a s y m p totic properties of s t a t i s t i c a l e s t i m a t i o n a n d t e s t p r o c e d u r e s , t h a t is, on p r o p .

.

erties whisk are possessed only in the limiting ca~e of samples of infinite size. H e c o n s t r u c t s two e x a m p l e s , one r e l a t i n g to t h e m e a n of a n o r m a l d i s t r i b u t i o n a n d t h e o t h e r to t h e e x t e n t of t h e o n e - p a r a m e t e r u n i f o r m d i s t r i b u t i o n , w h i c h show t h a t of two t e s t s w h i c h a r e asymcptoticcdly equivalent a s defined b y Neym a n a n d b o t h asymptotically " b e s t " in e i t h e r of two senses ( " a s y m p t o t i c a l l y m o s t p o w e r f u l " a n d " a s y m p t o t i c a l l y m o s t s t r i n g e n t , " a s defined b y W a l d ) , one *1~. L. Hsu. The limiting distribution of a general class of at~t~ttcs. demi& Sinica), 1942, 1, 87-41.

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m a y be universally better than the other, i.e., more powerful with respect to every member of the class of alternatives (to the null hypothesis) considered, in samples of any finite size. H e concludes: "Actually it seems doubtful that any definition of optimum tests, based only on asymptotic properties of power functions, can be very satisfactory, since in practice the sample size is always limited and since obviously an asymptotic property implies nothing about the behavior of any finite segment of a sequence of power functions." This conclusion undoubtedly carries over with equal force to any definition of optimum estimators involving only asymptotic properties, e.g., B A N estimates, regarding which Professor N e y m a n remarks: "The question remains open concerning h o w good these estimates are when the number of observations is only moderate." In this connection it needs to be emphasized that the method of m a x i m u m likelihood as developed by R. A. Fisher is not supported solely by the asymptotic properties of the estimators to which it leads, but has also to its credit that when ~ f f i c ~ t statistics (I prefer the term "exhaustive estimators") exist - - a finite sample property - - the method of maximum likelihood will lead to them. There are fashions in probability and statistics, j u s t as there are fashions in everyday affairs. A short while ago sequential analysis was the rage in statistics; a little later, the theory of games of strategy became the fad; and now the limelight seems to have shifted to stochastic processes and the fundamental mathematics of time se~cs. The papers on these two subjects in the P~'ocs~dings, by Feller and Doob, respectively, are therefore very timely now, several years a f t e r t h ei r original presentation in the Symposium. F o r most people the best way to gain an understanding of a new subject is not .by study of a formal tr e a t m e n t of it, but by analysis of dozens, or hundreds, of examples. F o r such people - - the present reviewer included - - Professor Feller's brilliant and lucid survey of stochastic processes from the viewpoint of applications, "On the Theory of Stochastic Processes, with P a r t i c u l a r Reference to Applications," is precisely what is needed. His expository paper on stochastic processes is certainly one of the best, and from many standpoints probably ~he best paper in the Proceedings volume. Professor Doob's paper, "Time Series and Harmonic Analysis," is an expository t r e a t m e n t of the fundamental mathematics of time series and harmonic analysis, and picks up the study of processes which evolve in time, space, or both, more or less where Professor Feller leaves off. The need for, and the aim of, this continuation is well expressed by Professor Doob in his opening paragraph: Although many articles on the present subject have appeared in the mathematical, statistical, and physical literature, there still seems to be some justification for one more. The statisticians have applied only small parts of the theory; the physicists have gone deeper, but write like physicists; the mathematicians have gone furthest, but write like mathematicians, only for posterity. Their work is frequently not understood, and is in general either ignored or applied in simplified forms which often are formally more formidable than the original rigorous one. The present paper attempts to give a compact outline of the harmonic analysis of stochastic processes, with applications to physical problems. While Professor Doob's paper is for the most p a r t "given over to the harmonic analysis of stochastic processes, the harmonic analysis of [non-stochastic] functions is outlined briefly, in order to exhibit the parallelism between the two . . . . and the g r e a t e r simplicity of the first." Although some applications to problems in physics, electrical circuit theory, and the like, are given, the exposition is quite mathematical and considerably more difficult to follow t h a n Professor Feller's treatment of the more elementary types of stochastic nrocesses. Before leaving the subject of time series, it should be noted t h a t t h e P~oceedi~gs contains a paper by G. F. McEwen on statistical procedures f o r tesr~ ing, "The Reality of Regularities Indicated in Sequences of Observations." Pro-

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lessor MeEwen's treatment of this subject is, however, fairly cursory and based on the literature of the subject up to 19~0, since which time there has been much activity and m a n y publications in this field. I n time it may be possible to assess the full influence of the Berkeley Symposium as a stimulus to "the r e t u r n to theoretical research" in probability and mathematical statistics. But such a n evaluation will be complicated by the fact that while the participants in the Symposium were "stimulated" in 1945-1946, other workers in probability and statistics, with the exception, perhaps, of students and immediate colleagues of the participants, had to wait until 1949 three years later - - to ,be "stimulated" by it and were not idle while they waited.

National Bureau of Standards

Churchill Eisenhart

BOOKS R E C E I V E D S. HOWARDBARTLEY. Beginning Ezpe~'i~nentat Psychology. New York: McGrawHill Book Co., 1950. Pp. 483 -~- vii. RAYMOND B. CATTELL. Persort~[~L New York: McGraw-Hill Book Co., 1950. Pp. 689 ~- xii. W. EDWARDS DEMING. So~rte Theo~t of Sa~rtpl~ng. New York: John Wiley & Sons, Inc., 1950. Pp. 602 ~-xvii. JAMES G. MILLI~. Ezpe~ments in Soc~l Process. New York: McGraw-Hill Book Co., 1950. Pp. 205 ~ ix. CLIFFORD T. MORGANAND ELIOT STELLAR. Physiological Psychology (2nd edition). New York: McGraw-Hill Book Co., 1950. Pp. 609 ~ ix. PHILIP E. VEI~ON AND JOHN B. PARRY. Personnel Selection in th~ B~'itish Forces. London: University of London Press, 1950. Pp. 324.