B.L. van der Waerden

Mathematical Statistics

With 39 Figures

1969

George Allen & Unwin Ltd. London

Springer-Verlag Berlin Heidelberg GmbH

Geschaftsfuhrende Herausgeber:

Prof. Dr. B. Eckmann Eidgenossische Technische Hochschule Zurich

Prof. Dr. B. L. van der Waerden Mathematisches Institut der Universitat Zur ich

Translation of

Mathematische Statistik

(Grundlehren der mathematischen Wissenschaften,

Bd. 87, lAuflage, 1965)

I S B N 9 7 8 - 3 - 6 6 2 - 2 2 1 3 9 - 6 I S B N 9 7 8 - 3 - 6 6 2 - 2 2 1 3 7 - 2 ( e B o o k )

D O I 10 .1007 /978 -3 -662 -22137 -2

This work is subject to copyright. AH rights are reserved, whether the whole o r part of the material is concerned, specifically those o f translation, repr int ing, re-use of i l lustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 o f the German Copyr ight Law where copies are made for other than private use, a fee is payable to the publisher, the amount o f the fee to be determined by agreement wi th the publisher. L ibrary o f Congress Catalog Card Number 72-84145.

© by Springer-Verlag Ber l in Heidelberg 1969 Orig inal ly published by Springer-Verlag Ber l in Heidelberg N e w York i n 1969 Softcover reprint o f the hardcover 1st edit ion 1969

T i t le N o . 5139

B. L. van der Waerden

Mathematical Statistics

With 39 Figures

Springer-Verlag Berlin Heidelberg GmbH 1969

Geschliftsftihrende Hel"dusgeber:

Prof. Dr. B. Eckmann Eidgenossische Techniscbe Hochschule Zurich

Prof. Dr. B. L. van der Waerden Mathematisches Institut der Universitlit Zurich

Translation of

Mathematische Statistik

(Grundlehren der mathematischen Wissenschaften,

Bd. 87, 2. Auflage, 1965)

ISBN 978-3-662-22139-6 ISBN 978-3-662-22137-2 (eBook) DOl 10.1007/978-3-662-22137-2

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concemed,specifieally those of translation, reprinting, rc-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisber, the amount of the fee to be determined by agreement with the publisher. Library of Congress Catalog Card Number 72-84145.

© by Springer-Verlag Berlin HeidelbeJg 1969 Original1ypublished by Springer-Verlag Berlin HeidelbeIgNew York in 1969 Softcover reprint of the hardcover 1st edition 1969

Title No. 5139

Die Grundlehren der mathematischen Wissenschaften

in Einzeldarstellungen mit besonderer Beriicksichtigung

der Anwendungsgebiete

Band 156

Herausgegeben von

J. L. Doob . A. Grothendieck . E. Heinz· F. Hirzebruch E. Hopf . H. Hopf . W. Maak . S. MacLane . W. Magnus M.M.Postnikov . F.K.Schmidt . D.S.Scott . K.Stein

Geschiiflsfiihrende H erausgeber

B. Eckmann und B. L. van der Waerden

B.L.van derWaerden

Mathematical Statistics

With 39 Figures

Springer-Verlag Berlin Heidelberg GmbH 1969

Geschliftsftihrende Hel"dusgeber:

Prof. Dr. B. Eckmann Eidgenossische Techniscbe Hochschule Zurich

Prof. Dr. B. L. van der Waerden Mathematisches Institut der Universitlit Zurich

Translation of

Mathematische Statistik

(Grundlehren der mathematischen Wissenschaften,

Bd. 87, 2. Auflage, 1965)

ISBN 978-3-662-22139-6 ISBN 978-3-662-22137-2 (eBook) DOl 10.1007/978-3-662-22137-2

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concemed,specifieally those of translation, reprinting, rc-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisber, the amount of the fee to be determined by agreement with the publisher. Library of Congress Catalog Card Number 72-84145.

© by Springer-Verlag Berlin HeidelbeJg 1969 Original1ypublished by Springer-Verlag Berlin HeidelbeIgNew York in 1969 Softcover reprint of the hardcover 1st edition 1969

Title No. 5139

Foreword to the English Edition

Chapters 1 - 7 were translated by Mrs. Virginia Thompson, and Chapters 8 -14 by Miss Ellen Sherman, at the University of California, Berkeley, and the University of Hull, England. In my opinion, both translators have done a superb job. In particular, they have taken care to find the correct translations of technical terms in widely divergent fields like biology and economics. They also have corrected several errors in formulae. They asked me every time they did not fully under­stand the reasoning. Several points are now explained more clearly than in the original German text.

Thanks are due to E. L. Lehmann (Berkeley), who initiated the trans­lation project and whose help was very valuable. Following his advice, I have added to § 4 a subsection on testing the mean of a normal distri­bution with given variance.

Zurich, ] uly 1968 B. L. van der Waerden

Foreword

Ever since my days as a student, economists, doctors, physiologists, biologists, and engineers have come to me with queries of a statistical nature. This book is the product of my long interest in practical solutions to such problems. Study of the literature and my own ideas have repeat­edly led me to improved methods, which will be established here and applied to instructive examples taken from the natural and social sciences. Thus, I hope to help the reader avoid the many fruitless direc­tions in which I worked at first. The examples are not artificially con­structed from theoretical considerations but are instead taken from real situations; consequently, many of the examples require detailed explana­tion.

The presentation of the basic mathematical concepts is, I hope, as brief as possible without becoming incomprehensible. Some rather long theoretical arguments have been necessary, but, whenever possible, references for the more difficult proofs have been made to good text­books already in existence. There would be no point in developing again the mathematical theories which have been presented clearly and in detail by Kolmogorov, Caratheodory, and Cramer.

A knowledge of the elements of the theory of functions and of the Lebesgue theory of integration has been assumed. This does not imply that a reader without such knowledge will not be able to understand the book: he will have either to accept certain theorems without proof or to confine himself to the more elementary sections, in which only analytic geometry and calculus are used (Chapters 1 through 4, 10, and 12).

The book is only meant to be an introduction. No attempt has been made at completeness, and such important topics as sequential analysis, decision theory, and stochastic processes have had to be omitted. How­ever, distinguished experts have written specialized books devoted to such topics:

A.Wald, Sequential Analysis. New York: JohnWiley&Sons 1947; A. Wald, Statistical Decision Functions. New York: John Wiley & Sons

1950; J. L. Doob, Stochastic Processes. New York: John Wiley & Sons 1953. In many places references for further reading have been given. These

have been placed where they are readily available, in the text or as

Foreword VII

footnotes. The new style of putting footnotes at the end of the book or, even worse, at the ends of the chapters causes a horrible amount of flipping back and forth through the pages. I also consider it preferable to write p. 5 rather than just 5. No attempt has been made at uniformity in quotations and extensive abbreviations have been avoided.

The first draft of this book was written in 1945 and served as the basis for a course on error theory and statistics at the Shell Laboratory in Amsterdam. A later version was read critically by Dr. E. Batschelet (Basel). I wish to express my thanks to him and also to Professor E. L. Lehmann (Berkeley) for their most valuable criticism. Also I thank Mr. H. R. Fischer and Mr. E. Nievergelt (Zurich) for drawing the figures and for their help with the proofreading.

September 1956 B. L. van der Waerden

Foreword to the Second Edition

In the new edition, Fig. 28, which was in error, has been redrawn by Mr. H. Studer.

April 1965 B. L. van der Waerden

Contents

Introduction . . . .

Chapter One: General Foundations

§ 1. Foundations of Probability Theory . . . . . § 2. Random Variables and Distribution Functions . . . § 3. Expectation and Standard Deviation . . . . . . . § 4. Integral Representation for Expectations and Probabilities

Chapter Two: Probabilities and Frequencies

§ 5. Binomial Distribution. . . . . . . . . . . § 6. Deviation of the Frequency h from the Probability p . § 7. Confidence Bounds for Unknown Probabilities § 8. Sampling . . . . . . . . . . . § 9. Comparison of Two Probabilities.

§ 10. Frequency of Rare Events . . . .

Chapter Three: Mathematical Tools

§ 11. Multiple Integrals. Transformations to Polar Coordinates § 12. Beta and Gamma Functions § 13. Orthogonal Transformations. . . § 14. Quadratic Forms and Their Invariants.

Chapter Four: Empirical Determination of Distribution Functions, Expectations, and Standard Deviations

§ 15. The Quetelet Curve. . . . . . . . . . . § 16. Empirical Determination of Distribution Functions § 17. Order Statistics . . . . . . . . . § 18. Sample Mean and Sample Variance. . § 19. Sheppard's Correction . . . . . . . § 20. Other Mean and Dispersion Measures.

Chapter Five: Fourier Integrals and Limit Theorems

§ 21. Characteristic Functions. § 22. Examples . . . . . § 23. The X2 Distribution. . . § 24. Limit Theorems . . . . § 25. Rectangular Distribution. Rounding Errors

3 8

12 17

24 27 32 37 40 47

52 55 60 61

67 69 75 79 82 85

89 93 95 97

104

x Contents

Chapter Six: Gauss Theory of Errors and Student's Test § 26. Gauss Theory of Errors § 27. The Distribution of S2. . . § 28. Student's Test . . . . . . § 29. Comparison of Two Means

Chapter Seven: Method of Least Squares § 30. Smoothing Observational Errors . . . . . . . . . . . § 31. Expectations and Standard Deviations of the Estimates [) . § 32. Estimation of the Variance (J2 . . . . . . . . . . . . § 33. Linear Regression . . . . . . . . . . . . . . . . . § 34. Causal Interpretation of Dependence between Economic Variables

Chapter Eight: Estimation of Unknown Constants § 35. R. A. Fisher's Method of Maximum Likelihood § 36. Determination of the Maximum . § 37. An Inequality Due to Frechet . . § 38. Sufficiency and Minimum Variance § 39. Examples . . . . . . . § 40. Conditional Expectation. . . . . § 41. Sufficient Statistics . . . . . . . § 42. Application to the Problem of Unbiased Estimation. § 43. Applications. . . . . . . . . . . . . . . . . . § 44. Estimation of the Variance of a Normal Distribution § 45. Asymptotic Properties . . . . . . . . . . . . .

Chapter Nine: Inferences Based on Observed Frequencies § 46. The Maximum Likelihood Method . . . . . . . . . § 47. Consistency of the Maximum Likelihood Estimate . . §48. Maximum Likelihood, Minimum /, and Least Squares § 49. Asymptotic Distributions of x2 and .9 § 50. Efficiency . § 51. The x2-Test . . . . . . . . . . .

Chapter Ten: Bio-Assay § 52. Response Curves and Logarithmic Response Curves. § 53. Integral-Approximation Method of Behrens and Karber § 54. Methods Based on the Normality Assumption § 55. "Up and Down" Methods. . . . . . . . . . . . .

Chapter Eleven: Hypothesis Testing § 56. Applications of the X2-Test. . . . § 57. The Variance-Ratio Test (F-Test). . . . § 58. The Analysis of Variance . . . . . . . § 59. General Principles. Most Powerful Tests. § 60. Composite Hypotheses . . . . . . . .

108 113 118 121

127 133 139 144 149

151 155 160 162 165 168 170 173 175 179 182

185 189 192 197 203 207

212 214 217 221

225 242 246 256 263

Contents

Chapter Twelve: Order Tests § 61. The Sign Test . . § 62. The Two-Sample Problem. . . § 63. Wilcoxon's Test . . . . . . . § 64. The Power of the Wilcoxon Test §65. The X-Test ........ .

Chapter Thirteen: Correlation § 66. Covariance and the Correlation Coefficient. . . . . . . § 67. The Correlation Coefficient as a Characteristic of Dependence § 68. Partial Correlation Coefficients. . . . . . . . . . . . § 69. Distribution of the Coefficient r for Dependent Variables. § 70. Spearman's Rank Correlation R § 71. Kendall's Rank Correlation T . . . . . . . . . . .

Chapter Fourteen: Tables

XI

267 271 273 282 290

301 305 310 316 323 333

Tables 1 -13 . . . . . .......... 339

Examples, Arranged According to Subject Matter Author and Subject Index . . . . . . . . . .

359 361