A History Of Mathematical Statistics From 1750 To 1930

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About A History Of Mathematical Statistics From 1750 To 1930

The long-awaited second volume of Anders Hald’s history of the development of mathematical statistics.
Anders Hald’s A History of Probability and Statistics and Their Applications before 1750 is already considered a classic by many mathematicians and historians. This new volume picks up where its predecessor left off, describing the contemporaneous development and interaction of four topics: direct probability theory and sampling distributions; inverse probability by Bayes and Laplace; the method of least squares and the central limit theorem; and selected topics in estimation theory after 1830.
In this rich and detailed work, Hald carefully traces the history of parametric statistical inference, the development of the corresponding mathematical methods, and some typical applications. Not surprisingly, the ideas, concepts, methods, and results of Laplace, Gauss, and Fisher dominate his account. In particular, Hald analyzes the work and interactions of Laplace and Gauss and describes their contributions to modern theory. Hald also offers a great deal of new material on the history of the period and enhances our understanding of both the controversies and continuities that developed between the different schools. To enable readers to compare the contributions of various historical figures, Professor Hald has rewritten the original papers in a uniform modern terminology and notation, while leaving the ideas unchanged.
Statisticians, probabilists, actuaries, mathematicians, historians of science, and advanced students will find absorbing reading in the author’s insightful description of important problems and how they gradually moved toward solution.

Table Of Content Of A History Of Mathematical Statistics From 1750 To 1930

Preface XV

  1. Plan of the Book 1
    1.1. Outline of the Contents
    1 1.2. Terminology and Notation 7
    1.3. Biographies 8
    PART I DIRECT PROBABILITY, 1750-1805
  2. Some Results and Tools in Probability Theory by Bernoulli, de Moivre, and Laplace 11
    2.1. The Discrete Equiprobability Model 11
    2.2. The Theorems of James and Nicholas Bernoulli, 1713 13
    2.3. The Normal Distribution as Approximation to the Binomial. De Moivre’s Theorem, 1733, and Its Modifications by Lagrange, 1776, and Laplace, 1812 17
    2.4. Laplace’s Analytical Probability Theory 25
  3. The Distribution of the Arithmetic Mean, 1756-1781 33
    3.1. The Measurement Error Model 33
    3.2. The Distribution of the Sum of the Number of Points by n Throws of a Die by Montmort and de Moivre 34
    3.3. The Mean of Triangularly Distributed Errors. Simpson, 1756-1757 35
    3.4. The Mean of Multinomially and Continuously Distributed Errors, and the Asymptotic Normality of the Multinomial. Lagrange, 1776 40
    3.5. The Mean of Continuous Rectangularly Distributed Observations. Laplace, 1776 50
    3.6. Laplace’s Convolution Formula for the Distribution of a Sum, 1781 55
  4. Chance or Design. Tests of Significance 65
    4.1. Moral Impossibility and Statistical Significance 65
    4.2. Daniel Bernoulli’s Test for the Random Distribution of the Inclinations of the Planetary Orbits, 1735 68
    4.3. John Michell’s Test for the Random Distribution of the Positions of the Fixed Stars, 1767 70
    4.4. Laplace’s Test of Significance for the Mean Inclination, 1776 and 1812 74
  5. Theory of Errors and Methods of Estimation 79
    5.1. Theory of Errors and the Method of Maximum Likelihood by Lambert, 1760 and 1765 79
    5.2. Theory of Errors and the Method of Maximum Likelihood by Daniel Bernoulli, 1778 83
    5.3. Methods of Estimation by Laplace before 1805 87
  6. Fitting of Equations to Data, 1750-1805 91
    6.1. The Multiparameter Measurement Error Model 91
    6.2. The Method of Averages by Tobias Mayer, 1750 94
    6.3. The Method of Least Absolute Deviations by Boscovich, 1757 and 1760 97
    6.4. Numerical and Graphical Curve Fitting by Lambert, 1765 and 1772 103
    6.5. Laplace’s Generalization of Mayer’s Method, 1787 107
    6.6. Minimizing the Largest Absolute Residual. Laplace, 1786, 1793, and 1799 108
    6.7. Laplace’s Modification of Boscovich’s Method, 1799 112
    6.8. Laplace’s Determination of the Standard Meter, 1799 116
    6.9. Legendre’s Method of Least Squares, 1805 118
    PART II INVERSE PROBABILITY BY BAYES AND LAPLACE, WITH COMMENTS ON LATER DEVELOPMENTS
  7. Induction and Probability: The Philosophical Background 125
    7.1. Newton’s Inductive-Deductive Method 125
    7.2. Hume’s Ideas on Induction and Probability, 1739 126
    7.3. Hartley on Direct and Inverse Probability, 1749 129
  8. Bayes, Price, and the Essay, 1764-1765 133
    8.1. Lives of Bayes and Price 133
    8.2. Bayes’s Probability Theory 136
    8.3. The Posterior Distribution of the Probability of Success 138
    8.4. Bayes’s Scholium and His Conclusion 142
    8.5. Price’s Commentary 145
    8.6. Evaluations of the Beta Probability Integral by
    Bayes and Price 147
  9. Equiprobability, Equipossibility, and Inverse Probability 155
    9.1. Bernoulli’s Concepts of Probability, 1713 155
    9.2. Laplace’s Definitions of Equiprobability and Equipossibility, 1774 and 1776 157
    9.3. Laplace’s Principle of Inverse Probability, 1774 159
    9.4. Laplace’s Proofs of Bayes’s Theorem, 1781 and 1786 164
  10. Laplace’s Applications of the Principle of Inverse
    Probability in 1774 167
    10.1. Introduction 167
    10.2. Testing a Simple Hypothesis against a Simple Alternative 167
    10.3. Estimation and Prediction from a Binomial Sample 169
    10.4. A Principle of Estimation and Its Application to Estimate the Location Parameter in the Measurement Error Model 171
    10.5. Laplace’s Two Error Distributions 176
    10.6. The Posterior Median Equals the Arithmetic Mean for a Uniform Error Distribution, 1781 180
    10.7. The Posterior Median for Multinomially Distributed Errors and the Rule of Succession, 1781 181
  11. Laplace’s General Theory of Inverse Probability 185
    11.1. The Memoirs from 1781 and 1786 185
    11.2. The Discrete Version of Laplace’s Theory 185
    11.3. The Continuous Version of Laplace’s Theory 188
  12. The Equiprobability Model and the Inverse Probability Model for Games of Chance 191
    12.1. Theoretical and Empirical Analyses of Games of Chance 191
    12.2. The Binomial Case Illustrated by Coin Tossings 192
    12.3. A Solution of the Problem of Points for Unknown Probability of Success 196
    12.4. The Multinomial Case Illustrated by Dice Throwing 197
    12.5. Poisson’s Analysis of Buffon’s Coin-Tossing Data 198
    12.6. Pearson and Fisher’s Analyses of Weldon’s Dice-Throwing Data 200
    12.7. Some Modem Uses of the Equiprobability Model 201
  13. Laplace’s Methods of Asymptotic Expansion, 1781 and 1785 203
    13.1. Motivation and Some General Remarks 203
    13.2. Laplace’s Expansions of the Normal Probability Integral 206
    13.3. The Tail Probability Expansion 210
    13.4. The Expansion about the Mode 212
    13.5. Two Related Expansions from the 1960s 216
    13.6. Expansions of Multiple Integrals 218
    13.7. Asymptotic Expansion of the Tail Probability of a Discrete Distribution 220
    13.8. Laplace Transforms 222
  14. Laplace’s Analysis of Binomially Distributed Observations 229
    14.1. Notation 229
    14.2. Background for the Problem and the Data 230
    14.3. A Test for the Hypothesis 6 _< r Against 0 > r Based on the Tail Probability Expansion, 1781 232
    14.4. A Test for the Hypothesis 0:< r Against 0 > r Based on the Normal Probability Expansion, 1786 234
    14.5. Tests for the Hypothesis 02< 01 Against 02 > 01, 1781, 1786, and 1812 235
    14.6. Looking for Assignable Causes 240
    14.7. The Posterior Distribution of 0 Based on Compound Events, 1812 242
    14.8. Commentaries 245
  15. Laplace’s Theory of Statistical Prediction 249
    15.1. The Prediction Formula 249
    15.2. Predicting the Outcome of a Second Binomial Sample from the Outcome of the First 249
    15.3. Laplace’s Rule of Succession 256
    15.4. Theory of Prediction for a Finite Population. Prevost and Lhuilier, 1799 262
    15.5. Laplace’s Asymptotic Theory of Statistical Prediction, 1786 264
    15.6. Notes on the History of the Indifference Principle and the Rule of Succession from Laplace to Jeffreys (1948) 268
  16. Laplace’s Sample Survey of the Population of France and the Distribution of the Ratio Estimator 283
    16.1. The Ratio Estimator 283 16.2. Distribution of the Ratio Estimator, 1786 284
    16.3. Sample Survey of the French Population in 1802 286
    16.4. From Laplace to Bowley (1926), Pearson (1928), and Neyman (1934) 289
    PART III THE NORMAL DISTRIBUTION, THE METHOD OF LEAST SQUARES, AND THE CENTRAL LIMIT THEOREM. GAUSS AND LAPLACE, 1809-1828
  17. Early History of the Central Limit Theorem, 1810-1853 303
    17.1. The Characteristic Function and the Inversion Formula for a Discrete Distribution by Laplace, 1785 303
    17.2. Laplace’s Central Limit Theorem, 1810 and 1812 307
    17.3. Poisson’s Proofs, 1824, 1829, and 1837 317
    17.4. Bessel’s Proof, 1838 327
    17.5. Cauchy’s Proofs, 1853 329
    17.6. Ellis’s Proof, 1844 333
    17.7. Notes on Later Developments 335
    17.8. Laplace’s Diffusion Model, 1811 337
    17.9. Gram-Charlier and Edgeworth Expansions 344
  18. Derivations of the Normal Distribution as a Law of Error 351
    18.1. Gauss’s Derivation of the Normal Distribution and the Method of Least Squares, 1809 351
    18.2. Laplace’s Large-Sample Justification of the Method of Least Squares and His Criticism of Gauss, 1810 357
    18.3. Bessel’s Comparison of Empirical Error Distributions with the Normal Distribution, 1818 360
    18.4. The Hypothesis of Elementary Errors by Hagen, 1837, and Bessel, 1838 365
    18.5. Derivations by Adrain, 1808, Herschel, 1850, and Maxwell, 1860 368
    18.6. Generalizations of Gauss’s Proof: The Exponential Family of Distributions 373
    18.7. Notes and References 380
  19. Gauss’s Linear Normal Model and the Method of Least Squares, 1809 and 1811 381
    19.1. The Linear Normal Model 381
    19.2. Gauss’s Method of Solving the Normal Equations 383
    19.3. The Posterior Distribution of the Parameters 386
    19.4. Gauss’s Remarks on Other Methods of Estimation 393
    19.5. The Priority Dispute between Legendre and Gauss 394
  20. Laplace’s Large-Sample Theory of Linear Estimation, 1811-1827 397
    20.1. Main Ideas in Laplace’s Theory of Linear Estimation, 1811-1812 397
    20.2. Notation 398
    20.3. The Best Linear Asymptotically Normal Estimate for One Parameter, 1811 399
    20.4. Asymptotic Normality of Sums of Powers of the Absolute Errors, 1812 401
    20.5. The Multivariate Normal as the Limiting Distribution of Linear Forms of Errors, 1811 402
    20.6. The Best Linear Asymptotically Normal Estimates for Two Parameters, 1811 405
    20.7. Laplace’s Orthogonalization of the Equations of Condition and the Asymptotic Distribution of the Best Linear Estimates in the Multiparameter Model, 1816 410
    20.8. The Posterior Distribution of the Mean and the Squared Precision for Normally Distributed Observations, 1818 and 1820 418
    20.9. Application in Geodesy and the Propagation of Error, 1818 and 1820 424
    20.10. Linear Estimation with Several Independent Sources of Error, 1820 430
    20.11. Tides of the Sea and the Atmosphere, 1797-1827 431 20.12. Asymptotic Efficiency of Some Methods of Estimation, 1818 444
    20.13. Asymptotic Equivalence of Statistical Inference byDirect and Inverse Probability 452
  21. Gauss’s Theory of Linear Unbiased Minimum Variance Estimation, 1823-1828 455
    21.1. Asymptotic Relative Efficiency of Some Estimates of the Standard Deviation in the Normal Distribution, 1816 455
    21.2. Expectation, Variance, and Covariance of Functions of Random Variables, 1823 459
    21.3. Gauss’s Lower Bound for the Concentration of the Probability Mass in a Unimodal Distribution, 1823 462
    21.4. Gauss’s Theory of Linear Minimum Variance Estimation, 1821 and 1823 465
    21.5. The Theorem on the Linear Unbiased Minimum Variance Estimate, 1823 467
    21.6. The Best Estimate of a Linear Function of the Parameters, 1823 476
    21.7. The Unbiased Estimate of a2 and Its Variance, 1823 477
    21.8. Recursive Updating of the Estimates by an Additional Observation, 1823 480
    21.9. Estimation under Linear Constraints, 1828 484
    21.10. A Review 488
    PART IV SELECTED TOPICS IN ESTIMATION THEORY, 1830-1930
  22. On Error and Estimation Theory, 1830-1890 493
    22.1. Bibliographies on the Method of Least Squares 493
    22.2. State of Estimation Theory around 1830 494
    22.3. Discussions on the Method of Least Squares and Some Alternatives 496
  23. Bienaymé’s Proof of the Multivariate Central Limit Theorem and His Defense of Laplace’s Theory of Linear Estimation, 1852 and 1853 501
    23.1. The Multivariate Central Limit Theorem, 1852 501
    23.2. Bravais’s Confidence Ellipsoids, 1846 504
    23.3. Bienaymé’s Confidence Ellipsoids and the x2 Distribution, 1852 506
    23.4. Bienaymé’s Criticism of Gauss, 1853 509
    23.5. The Bienaymé Inequality, 1853 510
  24. Cauchy’s Method for Determining the Number of Terms To Be Included in the Linear Model and for Estimating the Parameters, 1835-1853 511
    24.1. The Problem 511
    24.2. Solving the Problem by Means of the Instrumental Variable +1, 1835 512
    24.3. Cauchy’s Two-Factor Multiplicative Model, 1835 516
    24.4. The Cauchy-Bienaymé Dispute on the Validity of the Method of Least Squares, 1853 520
  25. Orthogonalization and Polynomial Regression 523
    25.1. Orthogonal Polynomials Derived by Laplacean Orthogonalization 523
    25.2. Chebyshev’s Orthogonal Polynomials, Least Squares, and Continued Fractions, 1855 and 1859 525
    25.3. Chebyshev’s Orthogonal Polynomials for Equidistant Arguments, 1864 and 1875 535
    25.4. Gram’s Derivation of Orthogonal Functions by the Method of Least Squares, 1879, 1883, and 1915 540
    25.5. Thiele’s Free Functions and His Orthogonalization of the Linear Model, 1889, 1897, and 1903 550
    25.6. Schmidt’s Orthogonalization Process, 1907 and 1908 556
    25.7. Notes on the Literature after 1920 on Least Squares Approximation by Orthogonal Polynomials with Equidistant Arguments 557
  26. Statistical Laws in the Social and Biological Sciences. Poisson, Quetelet, and Galton, 1830-1890 567
    26.1. Probability Theory in the Social Sciences by Condorcet and Laplace 567
    26.2. Poisson, Bienaymé, and Cournot on the Law of Large Numbers and Its Applications, 1830-1843 571
    26.3. Quetelet on the Average Man, 1835, and on the Variation around the Average, 1846 586
    26.4. Galton on Heredity, Regression, and Correlation, 1869-1890 599
    26.5. Notes on the Early History of Regression and Correlation, 1889-1907 616
  27. Sampling Distributions under Normality 633
    27.1. The Helmert Distribution, 1876, and Its Generalization to the Linear Model by Fisher, 1922 633
    27.2. The Distribution of the Mean Deviation by Helmert, 1876, and by Fisher, 1920 641
    27.3. Thiele’s Method of Estimation and the Canonical Form of the Linear Normal Model, 1889 and 1903 645
    27.4. Karl Pearson’s Chi-Squared Test of Goodnes of Fit, 1900, and Fisher’s Amendment, 1924 648
    27.5. “Student’s” t Distribution by Gosset, 1908 664
    27.6. Studentization, the F Distribution, and the Analysis of Variance by Fisher, 1922-1925 669
    27.7. The Distribution of the Correlation Coefficient, 1915, the Partial Correlation Coefficient, 1924, the Multiple Correlation Coefficient, 1928, and the Noncentral x2 and F Distributions, 1928, by Fisher 675
  28. Fisher’s Theory of Estimation, 1912-1935, and His Immediate Precursors 693
    28.1. Notation 693
    28.2. On the Probable Errors of Frequency Constants by Pearson and Filon, 1898 695
    28.3. On the Probable Errors of Frequency Constants by Edgeworth, 1908 and 1909 697
    28.4. On an Absolute Criterion for Fitting Frequency Curves by Fisher, 1912 707
    28.5. The Parametric Statistical Model, Sufficiency, and the Method of Maximum Likelihood. Fisher, 1922 713
    28.6. Efficiency and Loss of Information. Fisher, 1925 720
    28.7. Sufficiency, the Factorization Criterion, and the Exponential Family. Fisher, 1934 727
    28.8. Loss of Information by Using the Maximum Likelihood Estimate and Recovery of Information by Means ofAncillary Statistics. Fisher, 1925 729
    28.9. Examples of Ancillarity and Conditional Inference. Fisher, 1934 and 1935 732
    28.10. The Discussion of Fisher’s 1935 Paper 733
    28.11. A Note on Fisher and His Books on Statistics 734
    References 741

About The Author A History Of Mathematical Statistics From 1750 To 1930

ANDERS HALD, now retired, was Professor of Statistics at the University of Copenhagen from 1948 to 1982. His previous books include A History of Probability and Statistics and Their Applications before 1750, the companion volume to this history. He is an honorary Fellow of the Royal Statistical Society of London, the Danish Statistical Society, and the Danish Society for Industrial Quality Control.

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