Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability: Volume 3 Contributions to Astronomy and Physics [Reprint 2020 ed.] 9780520350694

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PROCEEDINGS OF T H I R D BERKELEY

THE

SYMPOSIUM

VOLUME I I I

PROCEEDINGS of the THIRD BERKELEY SYMPOSIUM ON MATHEMATICAL STATISTICS AND P R O B A B I L I T Y Held at the Statistical Laboratory University of California December, 1954 July and August, 1955

V O L U M E

III

CONTRIBUTIONS TO A S T R O N O M Y AND PHYSICS

EDITED BY J E R Z Y

UNIVERSITY

OF

N E Y M A N

C A L I F O R N I A

B E R K E L E Y A N D LOS A N G E L E S

1956

PRESS

U N I V E R S I T Y OF CALIFORNIA P R E S S B E R K E L E Y AND LOS A N G E L E S CALIFORNIA

CAMBRIDGE U N I V E R S I T Y P R E S S LONDON, ENGLAND

COPYRIGHT, 1 9 5 6 , BY T H E R E G E N T S O F T H E U N I V E R S I T Y OF CALIFORNIA

The United States Government and its officers, agents, and employees, acting within the scope of their duties, may reproduce, publish, and use this material in whole or in part for governmental purposes without payment of royalties thereon or therefor. The publication or republication by the government either separately or in a public document of any material in which copyright subsists shall not be taken to cause any abridgment or annulment of the copyright or to authorize any use or appropriation of such copyright material without the consent of the copyright proprietor.

LIBRARY OF CONGRESS CATALOG CARD NUMBER.: 4 9 - 8 1 8 9

P R I N T E D IN T H E U N I T E D STATES O F AMERICA

CONTENTS OF PROCEEDINGS, VOLUMES I, II, IV AND V Volume I—Statistics JOSEPH BERKSON, Estimation by least squares and by maximum likelihood. Z. W. BIRNBAUM, On the use of the Mann-Whitney statistic. HERMAN CHERNOFF and HERMAN RUBIN, The estimation of the location of a discontinuity in density. ARYEH DVORETZKY, On stochastic approximation. SYLVAIN EHRENFELD, Complete class theorems in experimental design. G. ELFVING, Selection of nonrepeatable observations for estimation. ULF GRENANDER and MURRAY ROSENBLATT, Some problems in estimating the spectrum of a time series. J. L. HODGES, JR. and E. L. LEHMANN, Two approximations to the RobbinsMonro process. WASSILY HOEFFDING, The role of assumptions in statistical decisions. SAMUEL KARLIN, Decision theory for Polya type distributions. L. LE CAM, On the asymptotic theory of estimation and testing hypotheses. HERBERT ROBBINS, An empirical Bayes approach to statistics. MURRAY ROSENBLATT, Some regression problems in time series analysis. CHARLES STEIN, Efficient nonparametric testing and estimation. CHARLES STEIN, Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. B. L. VAN D E R WAERDEN, The computation of the X-distribution.

Volume II—Probability Theory DAVID BLACKWELL, On a class of probability spaces. SALOMON BOCHNER, Stationarity, boundedness, almost periodicity of random-valued functions. K. L. CHUNG, Foundations of the theory of continuous parameter Markov chains. A. H. COPELAND, SR. Probabilities, observations and predictions. J. L. DOOB, Probability methods applied to the first boundary value problem. ROBERT FORTET, Random distributions with an application to telephone engineering. J. M. HAMMERSLEY, Zeros of a random polynomial. T. E. HARRIS, The existence of stationary measures for certain Markov processes. KIYOSI ITO, Isotropic random current. PAUL LEVY, A special problem of Brownian motion, and a general theory of Gaussian random functions. MICHEL LOEVE, Variational terms and central limit problem. EUGENE LUKACS, Characterization of populations by properties of suitable statistics. KARL MENGER, Random variables from the point of view of a general theory of variables. E D I T H MOURIER, L-Random elements and L*-random elements in Banach spaces. R. SALEM and A. ZYGMUND, A note on random trigonometric polynomials.

Volume IV—Biology and Problems of Health J. CROW and M. KIMURA, Some genetic problems in natural populations. E. R. DEMPSTER, Some genetic problems in controlled populations. JERZY NEYMAN, THOMAS PARK and ELIZABETH SCOTT, Struggle for existence. The Tribolium Model: biological and statistical aspects. M. S. BARTLETT, Deterministic and stochastic models for recurrent epidemics. A. T. BHARUCHA-REID, On the stochastic theory of epidemics. C. L. CHIANG, J. L. HODGES, JR., and J. YERUSHALMY, Statistical problems in medical diagnoses. JEROME CORNFIELD, A statistical problem arising from retrospective studies. D. G. KENDALL, Deterministic and stochastic epidemics in closed populations. W. F. TAYLOR, Problems in contagion.

Volume V—Econometrics, Industrial Research, and Psychometry K. J. ARROW and LEONID HURWICZ, Reduction of constrained maxima to saddle-point problems. E. W. BARANKIN, Toward an objectivistic theory of probability. C. W. CHURCHMAN, Problems of value measurement for a theory of induction and decisions. PATRICK SUPPES, The role of subjective probability and utility in decision making. A. H. BOWKER, Continuous sampling plans. CUTHBERT DANIEL, Fractional replication in industrial research. MILTON SOBEL, Sequential procedures for selecting the best exponential population. T. W. ANDERSON and HERMAN RUBIN, Statistical inference in factor analysis. FREDERICK MOSTELLER, Stochastic learning models. HERBERT SOLOMON, Probability and statistics in psychometric research: Item analysis and classification techniques.

ACKNOWLEDGMENT The astronomical papers printed in this volume were delivered at the two sessions on Astronomy, December, 1954, organized with the cooperation of Professors C. D. Shane and 0 . Struve. Most of the remaining papers were delivered at the session on Physics, also held in December, 1954, organized with the cooperation of Professor Mina Rees and Professor William A. Nierenberg acting, respectively, for Section A of the American Association for the Advancement of Science, and for the American Physical Society.

PREFACE T H E T H I R D B E R K E L E Y SYMPOSIUM on Mathematical Statistics and Probability was held in two parts, one from December 26 to 31, 1954, emphasizing applications, and the other, in July and August, 1955, emphasizing theory. The Symposium was thus divided because, on the one hand, it was thought desirable to provide an opport u n i t y for contacts between American and foreign scholars who could come to Berkeley in the summer, b u t not in the winter, and because, on the other hand, the 121st Annual Meeting of the American Association for the Advancement of Science held in Berkeley in December, 1954, provided an opportunity for joint sessions on the various fields of applications with its many member societies. W i t h the help of D r . Raymond L. Taylor, of the AAAS, nine cosponsored sessions of the Symposium were organized. Two of these were given to problems of astronomy and one each to biology, medicine and public health, statistical mechanics, industrial research, psychometry, philosophy of probability, and to statistics proper. T h e importance of the second part of the Symposium, which emphasized theory, was increased by the decision of the Council of the Institute of Mathematical Statistics to hold its first Summer Institute in Berkeley and to hold this Institute "in conjunction with the Third Berkeley Symposium"; all members of the I M S Summer Institute were invited to participate in the Symposium and the two enterprises were conducted in parallel. I n particular, the cooperation of Professor David Blackwell, Chairman of the I M S Summer Institute, made it possible to ensure t h a t representatives of all the major centers of statistical research in this country be invited. As will be seen from the lists of contents of the Proceedings, the response was good, although various circumstances, including the concurrent Rio de Janeiro meeting of the International Statistical Institute, prevented some of the prospective participants from attending the Berkeley meetings. Two months were allotted to the second part of the Symposium in order to provide an opportunity not only for formal presentation of papers, b u t also for informal contacts among the participants. To facilitate such personal associations, after three weeks of intensive lectures and discussions, a trip was made to the Sierra. There, animated discussions of stochastic processes and of decision functions were interspersed with expressions of delight at the beauty of Yosemite Valley, Emerald Bay, and Feather River Canyon. After this vacation there was another period of intensive lecturing. Although much effort was expended to arrange lectures and personal contacts, the primary concern of the Statistical Laboratory and of the D e p a r t m e n t of Statistics was with the Proceedings. Because of the participation of the AAAS, the amount of material submitted for publication was estimated to be equivalent to thirteen hundred printed pages, roughly twice the length of the Proceedings of the Second Berkeley Symposium. This presented a most embarrassing problem. T h a t it was finally solved is largely the result of the most effective support and advice of Dr. John Curtiss, Executive Director of the American Mathematical Society. His organizational talent and friendly help are greatly appreciated. Special t h a n k s are due Mr. August Frug6, the Manager of the Publishing Department of the University of California Press, and also his staff, who undertook the difficult and costly publication in the best spirit of cooperation with, and of service to, the scholarly community.

Vili

PREFACE

Since a single thirteen-hundred-page volume would have been difficult to handle and, for the majority of scholars, too expensive to buy, it was decided to issue the Proceedings in five relatively small volumes, each given to a specialized and, so far as possible, unified cycle of ideas. A list of contents of the other four volumes of the Proceedings will be found preceding this preface. The initial steps in the organization of the Symposium were based on a grant obtained from the University of California through the good offices of Professor Clark Kerr, Chancellor of the Berkeley campus of the University of California, to whom sincere thanks are due. This grant was followed by an appropriation from the Editorial Committee of the University of California, which provided the nucleus of the fund eventually collected for the publication of the Proceedings. This action of the Editorial Committee is gratefully acknowledged. For further effective support of the Symposium thanks must be given the National Science Foundation, the United States Air Force Research and Development Command, the United States Army Office of Ordnance Research, and the United States Navy Office of Naval Research. It is hoped that the material in the present Proceedings and, particularly, the scientific developments stimulated by the Symposium, will be sufficiently important to prove that the money received from these organizations was well spent. The success of the Symposium was, in large part, made possible by the generous and effective support of a number of scholarly societies. Sessions of the first part of the Symposium were sponsored by the American Physical Society; the American Statistical Association; the Astronomical Society of the Pacific; the Biometric Society, Western North American Region; the Ecological Society of America; the Institute of Mathematical Statistics; the Philosophy of Science Association; and the Western Psychological Society. The American Mathematical Society sponsored the second part of the Symposium, delegating for organizational work two of its most distinguished members, Professor J. L. Doob and Professor G. Polya, whose advice and cooperation were most helpful. The 1955 Summer Institute of the Institute of Mathematical Statistics was held in conjunction with the Symposium; the IMS also held its Western Regional Meeting in Berkeley in July. All papers published in these Proceedings were written at the invitation of the Statistical Laboratory, acting either on its own initiative or at the suggestion of the cooperating groups, and the Laboratory is, therefore, responsible for the selection of the authors; a responsibility that does not extend to the contents of the papers. The editorial work on the manuscripts submitted was limited to satisfying the requirements of the University of California Press regarding the external form of the material submitted, the numbering of formulas, etc., and to correcting obvious errors in transcription. Most of this was done by the staff of the Laboratory; in particular, Miss Catherine FitzGibbon, Mrs. Jeanne Lovasich, Mrs. Kathleen Wehner, and my colleague, Dr. Elizabeth L. Scott, who supervised some of the work. Occasionally, manuscripts were read by other participants in the Symposium particularly interested in them, and the authors were offered suggestions. However, in no case was there any pressure on the authors to introduce any material change into their work. Jerzy Neyman Director, Statistical Chairman, Department

Laboratory of Statistics

CONTENTS I. Contributions to Astronomy (i) Hertzsprung-Russell Diagram EGGEN—The Relationship between the Color and the Luminosity of Stars near the Sun

OLIN J.

L. GREENSTEIN—The Spectra and Other Properties of Stars Lying Below the Normal Main Sequence

1

JESSE

HAROLD L.

11

JOHNSON—Photoelectric Studies of Stellar Magnitudes and

Colors

31

E. KRON—Evidence for Sequences in the Color-Luminosity for M-Dwarfs

39

GERALD

BENGT

STRÔMGREN—The Hertzsprung-Russell Diagram

. . . .

49

(ii) Spatial Distribution of Galaxies G. C. MCVITTIE—Galaxies, Statistics and Relativity

69

N E Y M A N , E L I Z A B E T H L . S C O T T and C. D. SHANE—Statistics of Images of Galaxies with Particular Reference to Clustering . .

75

F. ZWICKY—Statistics of Clusters of Galaxies. Distribution of Centers, Angular Dimensions, Structure, Luminosity Function of Member Galaxies

113

JERZY

II. Contributions to Physics and and Probability Theory

ANDRÉ BLANC-LAPIERRE

ALBERT

TORTRAT—Statistical Mechanics

145

M. KAC—Foundations of Kinetic Theory J. KAMPÉ D E

171

FERIET—Random Solutions of Partial Differential

Equations

199

MONTROLL—Theory of the Vibration of Simple Cubic Lattices with Nearest Neighbor Interactions 209

ELLIOTT

NORBERT

WIENER—Nonlinear Prediction and Dynamics

. . . .

247

THE RELATIONSHIP BETWEEN THE COLOR AND LUMINOSITY OF STARS NEAR THE SUN OLIN J. EGGEN LICK OBSERVATORY, UNIVERSITY OF CALIFORNIA

The tabulation and description of all the individual stars was once considered to be the ultimate goal of astronomy. This Herculean task, which can no longer be seriously considered since the number of known stars and the variety of their physical and astrometric properties available for observations are so great, has been considerably lightened by the application of statistical methods. The observed quantities that we shall deal with here are VE The apparent, visual magnitude. (P — V)E The color or difference between the visual and the photographic magnitude. x(i) The trigonometric parallax. The values of ir(t) and VE are combined to give the absolute visual magnitude, MR = VE + 5 + 5 log™ IR(t) .

(1)

It is assumed that the nearby stars are not affected by interstellar absorption. The correlation to be investigated is that between Mv and (P — V)E and will be referred to as the color-luminosity array. The three samples of nearby stars to be discussed are characterized as follows, for (P — V)E < +l m 25: Group I

A-(i)

Weight

Mv

Number

Per cent observed

All

52

90

>0r050

>36

II

>0.050

16-36

20

+ 1 6 . In Luyten's diagram, figure 1, curves of constant diameter are shown, that is, the predictions of equation (1), and I wish to point out that the white dwarfs show a mean trend parallel to such lines. The scatter in luminosity at a given color, I then interpret as caused by the initial differences in mass and composition. (The abundance of the heavier elements affects the opacity, so that the rate of cooling is sensitive to composition). For a constant composition, location in the H-R diagram uniquely gives the age and the mass. Since only three masses have been actually measured, no test of this hypothesis is now possible. I have previously reported the existence of shells of ejected helium near the two helium-rich white dwarfs, and of Ca II near the F-type white dwarf, van Maanen 2 [6]. The spectroscopic results so far obtained at Palomar are summarized below. I t should be remembered that because of its very high speed, Bowen's 8-inch aplanatic camera has a spectral range limited to 670 A on one exposure. No data for the red-green region are available. 2.1. Group A. Normal hydrogen type. The best known and apparently simplest group proves to be that with strong broad hydrogen lines. Many objects are known, but spectrophotometric profiles for H7, H 0''30. The result is M = H — 2.5, with Mpg ranging from + 5 to + 7 . The two determinations of the M, H relation are not completely independent, but the mean result is (6)

M = H-2.9,

(sd and id),

26

THIRD BERKELEY SYMPOSIUM:

GREENSTEIN

disagreeing by about 0.9 mag. from my earlier result [19]. Equation (6) permits luminosity determinations from proper motions only, can be used for more stars, and seems to give somewhat better results than the few available parallaxes. In a few cases where n is accidentally very small,3 H cannot be used. Figures 6 and 7 show the results. It can be seen that there is no large deviation of the wk stars from the normal main sequence, and that the sd and id stars are below and to the left. +i +2

+3

— +4

x

E o £

X+5 A. Z

+6

+7

+8

+9

AO

A5

FO

F5

GO

G5

Ko

FIGURE 7

The H-R diagram based on proper motions, for the id and sd stars of the present investigation. (X stars classified by others, no line strength data available.) No wk stars are included. The colon is for a star of accidentally small proper motion. The G2 and G8 stars near M = +8.5 have extremely large proper motions; the G8 star is Groombridge 1830, with a trigonometric luminosity M „ = +7.4.

The mean values of Mv„ are given in table IV; there is a larger fraction of id's in the later types. The data for stars earlier than FO are poor. In the mean, the luminosity of the subdwarfs is about +1.4 mag. less than that on the main sequence for types later than FO; those I classify earlier may deviate slightly more. The gross mean Mvg for the range FO to GO is +5.4, or about M„ = +5.0. In this connection, if we use the value Mv = + 5 for the stars where Miss Roman obtained galactic orbits ([17], table III) we obtain what might be viewed as less extreme orbital characteristics. She finds that if M = + 4 , out of 13 orbits 7 are retrograde and 5 are hyperbolic; on the other hand, if M = + 5 , only 2 are retrograde and 3 are hyper3 For example H D 161817, idA3, has a radial velocity of - 3 6 3 km/sec, Ma = -0''064, M = 0''016 and is near the solar apex. An interesting sidelight on H D 161817, approaching us at high speed, with small transverse motion, is that it will eventually become one of the brightest stars in the sky. If it has M = + 2 , its distance is now 100 parsecs, its VT = 33 km/sec; it will be only 9 parsecs from the sun in about 280,000 A.D., and of apparent magnitude +1™8!

27

SPECTRA OF STARS

bolic. The distribution perpendicular to the galactic plane, in the latter case, is also more concentrated than for the RR Lyrae stars. If we used only luminosities derived from trigonometric parallaxes, 19 stars give Af(trig) - M(H) = + 0 . 6 mag. (One object +38°4955, sdF6 (Popper) is omitted because its M = +10.3 seems unreasonably large, although p = 0''052 ± 0T013). This difference is in the sense mentioned above, where two methods of obtaining relations between M and H differed by +0.8 mag. But the sense is such that, if only trigonometric parallaxes had been used, the sd and id stars would be + 2 . 0 mag. below the main sequence. Our conclusions as to their luminosity are therefore conservative. Both the spectral and luminosity analyses should be carried further. There is now good evidence that the sd and id stars can be distinguished from the ordinary wk TABLE

IV

Luminosities of S u b d w a r f s Type

No.

¿A5 A6-F0

5

F1-F5 F6-F9 G0-G8

6 13 10 9

Mpe from H sd + id

Main Sequence

+3.5 +5.2

+1.5 +2.4

+5.0

+3.6 +4.4

+6.3 +7.4

+5.4

AMPu and

m.e.

+2.0

±1.5

+2.8 +1.4

±0.9 ±0.4

+1.9

±0.3

+2.0

±0.4

high-velocity stars, both in luminosity and in weakening of the lines. We do not know, however, whether there is a continuous band of stars below the main sequence from 0 to + 3 mag. fainter, or whether there are discrete sequences. The wk stars need a similar but more elaborate analysis, and probably deviate less; stars like 10 CVn, r Cet and fi Cas belong to the wk group, and are slightly below the main sequence. 4.3. The velocity program. The Palomar subdwarf spectra yield remarkably accurate velocities. One plate containing 30 measurable lines gives an internal mean error of ± 0 . 6 to ±1.0 km/sec. Most subdwarf velocities, hitherto, had accuracies near ± 5 to ± 1 0 km/sec. Consequently, spectroscopic binaries can be found from very few observations. Since the main emphasis is on the spectral characteristics of these stars, two or more spectra will be obtained only for 40 of the 80 objects, repeating especially those whose velocities differed from published values [ 18]. Till now the only probable spectroscopic binary is —3°2525, (20C501), idF6, a 9 m 5 star. The published velocity of + 2 5 km/sec. included several discordant measures; my two plates give +53.5 ± 0.7 and +41.0 ± 0.6 km/sec. As yet I have no idea of the period. The luminosity is about + 6 T 0 . Photoelectric observations for possible eclipse would be very important, because a direct determination of radius and mass has never been made for an extreme subdwarf. A few other objects have velocities differing by 10 km/sec from published results, but the majority have differences nearer 5 km/sec. The number of velocity variables found will be small, and the percentage of binaries must also be small. One general fact is that all the A5-G8 subdwarfs have sharp lines, that is, small rotation; — 3°2525 was first noted as suspect because of a slight rotational broadening.

28

THIRD BERKELEY SYMPOSIUM! GREENSTEIN

4.4. Composition. Till now, the abundance determinations in high-velocity dwarfs are: three wk stars [20], where CH was found strong, the metals weak by small factors; two subdwarfs [21] in which a large reduction of the metallic abundances was indicated. L. H. Aller and the author are reanalyzing HD 19445, 140283, sdF5, and have added HD 161817, idA3 and HD 219617, sdGl; the author has investigated HD 103095, sdG8. The current interpretation of the high-velocity wk group is that it has perhaps one-half the population I metal abundance; the subdwarfs, sd and id, may have as little as one-twentieth the normal metal [21]. There is no evidence on the H / H e ratio. With the low luminosities now found, the F and G subdwarfs need not have appreciably changed their initial H / H e ratio by nuclear processes. The kinematical properties run with the division into wk, id and sd stars. It may be conjectured that the sd are the purest and oldest type II population, and have the lowest metal abundances; the id are quite similar if less extreme, and the wk stars are ordinary "high-velocity" stars, that is, possibly a mixture of disk populalation II and old population I, with nearly normal abundances. The first recent attempt to give a model for the internal structure of homogeneous stars with small heavy-element abundance has been made by Reiz [22]; opacity arises from free-free transitions of H and He and energy from the proton-proton chain. In spite of many theoretical uncertainties, it is interesting to note that his H-R diagram for objects near a solar mass ([22], figure 1), homologous stars built on his model, lies +1.0 to + 1.5 mag. below the main sequence and even further below for low hydrogen abundance. If the subdwarfs are in fact shown to be metal-poor by detailed spectroscopic analysis, there are at least two possible evolutionary modes. Since kinematically they are extreme population II stars, they may have been formed from gases, near the galactic center, in the early days of our galaxy, before heavy elements had appreciably evolved in supernovae, novae or white dwarfs [23]. The low metal content favors Hoyle's views, rather than those of Gamow and collaborators; on Gamow's view the heavy element content and interstellar gas of the stars has not essentially changed. However, the possibility remains that star formation in the galactic center region was from an initially metal-poor gas, while the population I stars are formed from gas and dust. The dust is substantially enriched in metals and in C, N, O, with respect to H and to He. The subdwarf problem is obviously related to the location of the main sequence of globular clusters. The giants in clusters show apparently low metal abundances, from spectra and from theoretical models. Baum states that in M 13 the so-called main sequence lies two magnitudes below the normal position [24]. -o-

-o

-c-

-O