A History of Mathematical Statistics from 1750 to 1930

Contents
Preface
1. Plan of the Book
	1.1 Outline of the Contents
	1.2 Terminology and Notation
	1.3 Biographies
PART I. Direct Probability 1750-1805
	2. Some Results and Tools in Probability Theory by Bernoulli, de Moivre, and Laplace
		2.1 The Discrete Equiprobability Model
		2.2 The Theorems of James and Nicholas Bernoulli, 1713
		2.3 The Normal Distribution as Approximation to The Binomial. De Moivre\'s Theorem, 1733, and Its Modifications by Lagrange, 1776, and Laplace, 1812
		2.4 Laplace\'s Analytical Probability Theory
	3. The Distribution of the Arithmetic Mean, 1756-1781
		3.1 The Measurement Error Model
		3.2 The Distribution of The Sum of The Number of Points by n Throws of a Die by Montmort and de Moivre
		3.3 The Mean of Triangularly Distributed Errors. Simpson, 1756-1757
		3.4 The Mean of Multinomially and Continuously Distributed Errors, and The Asymptotic Normality of The Multinomial. Lagrange, 1776
		3.5 The Mean of Continuous Rectangularly Distributed Observations. Laplace, 1776
		3.6 Laplace\'s Convolution Formula for the Distribution of a Sum, 1781
	4. Chance or Design. Tests of Significance
		4.1 Moral Impossibility and Statistical Significance
		4.2 Daniel Bernoulli\'s Test for the Random Distribution of the Inclinations of the Planetary Orbits, 1735
		4.3 John Michell\'s Test for the Random Distribution of The Positions of the Fixed Stars, 1767
		4.4 Laplace\'s Test of Significance for the Mean Inclination, 1776 and 1812
	5. Theory of Errors and Methods of Estimation
		5.1 Theory of Errors and the Method of Maximum Likelmood by Lambert, 1760 and 1765
		5.2 Theory of Errors and the Method of Maximum Likeldiood by Daniel Bernoulli, 1778
		5.3 Methods of Estimation by Laplace before 1805
	6. Fitting of Equations to Data, 1750-1805
		6.1 The Multiparameter Measurement Error Model
		6.2 The Mefflod of Averages by Tobias Mayer, 1750
		6.3 The Method of Least Absolute Deviations by Boscovich, 1757 and 1760
		6.4 Numerical and Grapmcal Curve Fitting by Lambert, 1765 and 1772
		6.5 Laplace\'s Generalization of Mayer\'s Method, 1787
		6.6 Minimizing the Largest Absolute Residual. Laplace, 1786, 1793, 1799
		6.7 Laplace\'s Modification of Boscovich\'s Method, 1799
		6.8 Laplace\'s Determination of the Standard Meter, 1799
		6.9 Legendre\'s Method of Least Squares, 1805
PART II. Inverse Probability by Bayes and Laplace, with Comments on Later Developments
	7. Induction and Probability: the Philosophical Background
		7.1 Newton\'s Inductive-deductive Method
		7.2 Hume\'s Ideas on Induction and Probability, 1739
		7.3 Hartley on Direct and Inverse Probability, 1749
	8. Bayes, Price, and the Essay, 1764-1765
		8.1 Lives of Bayes and Price
		8.2 Bayes\'s Probability Theory
		8.3 The Posterior Distribution of the Probability of Success
		8.4 Bayes\'s Scholium and His Conclusion
		8.5 Price\'s Commentary
		8.6 Evaluations of the Beta Probability Integral by Bayes and Price
	9. Equiprobability, Equipossibility, and Inverse Probability
		9.1 Bernoulli\'s Concepts of Probability, 1713
		9.2 Laplace\'s Definitions of Equiprobability and Equipossibility, 1774 and 1776
		9.3 Laplace\'s Principle of Inverse Probability, 1774
		9.4 Laplace\'s Proofs of Bayes\'s Theorem, 1781 and 1786
	10. Laplace\'s Applications of the Principle of Inverse Probability in 1774
		10.1 Introduction
		10.2 Testing a Simple Hypothesis against a Simple Alternative
		10.3 Estimation and Prediction from a Binomial Sample
		10.4 A Principle of Estimation and Its Application to Estimate the Location Parameter in the Measurement Error Model
		10.5 Laplace\'s Two Error Distributions
		10.6 The Posterior Median Equals the Arithmetic Mean for a Uniform Error Distribution, 1781
		10.7 The Posterior Median for Multinomially Distributed Errors and the Rule of Succession, 1781
	11. Laplace\'s General Theory of Inverse Probability
		11.1 The Memoirs from 1781 and 1786
		11.2 The Discrete Version of Laplace\'s Theory
		11.3 The Continuous Version of Laplace\'s Theory
	12. The Equiprobability Model and the Inverse Probability Model for Games of Chance
		12.1 Theoretical and Empirical Analyses of Games of Chance
		12.2 The Binomial Case Illustrated by Coin Tossings
		12.3 A Solution of the Problem of Points for Unknown Probability of Success
		12.4 The Multinomial Case Illustrated by Dice Throwing
		12.5 Poisson\'s Analysis of Buffon\'s Coin-tossing Data
		12.6 Pearson\'s and Fisher\'s Analyses of Weldon\'s Dice-throwing Data
		12.7 Some Modern Uses of the Equiprobability Model
	13. Laplace\'s Methods of Asymptotic Expansion, 1781 and 1785
		13.1 Morivation and Some General Remarks
		13.2 Laplace\'s Expansions of the Normal Probability Integral
		13.3 The Tail Probability Expansion
		13.4 The Expansion about the Mode
		13.5 Two Related Expansions from the 1960s
		13.6 Expansions of Multiple Integrals
		13.7 Asymptotic Expansion of the Tail Probability of a Discrete Distribution
		13.8 Laplace Transforms
	14. Laplace\'s Analysis of Binomially Distributed Observations
		14.1 Notation
		14.2 Background for the Problem and the Data
		14.3 A Test for the Hypothesis θ≦r against θ>r Based on the Tail Probability Expansion, 1781
		14.4 A Test for the Hypothesis θ≦r against θ>r Based on the Normal Probability Expansion, 1786
		14.5 Tests for the Hypothesis θ₂≦θ₁ against θ₂>θ₁, 1781, 1786, and 1812
		14.6 Looking for Assignable Causes
		14.7 The Posterior Distribution of θ Based on Compound Events, 1812
		14.8 Commentaries
	15. Laplace\'s Theory of Statistical Prediction
		15.1 The Prediction Formula
		15.2 Predicting the Outcome of a Second Binomial Sample from the Outcome of the First
		15.3 Laplace\'s Rule of Succession
		15.4 Theory of Prediction for a Finite Population. Prevost and Lhuilier, 1799
		15.5 Laplace\'s Asymptotic Theory of Statistical Prediction, 1786
		15.6 Notes on the History of the Indifference Principle and the Rule of Succession from Laplace to Jeffreys (1948)
	16. Laplace\'s Sample Survey of the Population of France and the Distribution of the Ratio Estimator
		16.1 The Ratio Estimator
		16.2 Distribution of the Ratio Estimator, 1786
		16.3 Sample Survey of the French Population in 1802
		16.4 From Laplace to Bowley (1926), Pearson (1928), and Neyman (1934)
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
	17.1 The Characteristic Function and the Inversion Formula for a Discrete Distribution by Laplace, 1785
	17.2 Laplace\'s Central Limit Theorem, 1810 and 1812
	17.3 Poisson\'s Proofs, 1824, 1829, and 1837
	17.4 Bessel\'s Proof, 1838
	17.5 Cauchy\'s Proofs, 1853
	17.6 Ellis\'s Proof, 1844
	17.7 Notes on Later Developments
	17.8 Laplace\'s Diffusion Model, 1811
	17.9 Gram-Charlier and Edgeworm Expansions
	Derivations of the Normal Distribution as a Law of Error
	18.1 Gauss\'s Derivation of the Normal Distribution and the Method of Least Squares, 1809
	18.2 Laplace\'s Large-sample Justification of the Method of Least Square and His Criticism of Gauss, 1810
	18.3 Bessel\'s Comparison of Empirical Error Distributions with the Normal Distribution, 1818
	18.4 The Hypothesis of Elementary Errors by Hagen, 1837, and Bessel, 1838
	18.5 Derivations by Adrain, 1808, Herschel, 1850, and Maxwell, 1860
	18.6 Generalizations of Gauss\'s Proof: the Exponential Family of Distributions
	18.7 Notes and References
	19. Gauss\'s Linear Normal Model and the Method of Least-Squares, 1809 and 1811
	19.1 The Linear Normal Model
	19.2 Gauss\'s Method of Solving the Normal Equations
	19.3 The Posterior Distribution of the Parameters
	19.4 Gauss\'s Remarks on Other Methods of Estimation
	19.5 The Priority Dispute between Legendre and Gauss
	Note to §19.3 added in Proof
	20. Laplace\'s Large-sample Theory of Linear Estimation, 1811-1827
	20.1 Main Ideas in Laplace\'s Theory of Linear Estimation, 1811-1812
	20.2 Notation
	20.3 The Best Linear Asymptotically Normal Estimate for One Parameter, 1811
	20.4 Asymptotic Normality of Sums of Powers of the Absolute Errors, 1812
	20.S the Multivariate Normal as the Limiting Distribution of Linear Forms of Errors, 1811
	20.6 the Best Linear Asymptotically Normal Estimates for Two Parameters, 1811
	20.7 Laplace\'s Orthogonalization of the Equations of Condmon and the Asymptotic Distribution of the Best Linear Estimates in the Multiparameter Model, 1816
	20.8 The Posterior Distribution of the Mean and the Squared Precision for Normally Distributed Observations, 1818 and 1820
	20.9 Applications in Geodesy and the Propagation of Error, 1818 and 1820
	20.10 Linear Estimation with Several Independent Sources of Error, 1820
	20.11 Tides of the Sea and the Atmosphere, 1797-1827
	20.12 Asymptotic Efflciency of Some Methods of Estimation, 1818
	20.13 Asymptotic Equivalence of Statistical Inference by Direct and Inverse Probability
	21. Gauss\'s Theory of Linear Unbiased Minimum Variance Estimation, 1823-1828
	21.1 Asymptotic Relative Efficiency of Some Estimates of the Standard Deviation in the Normal Distribution, 1816
	21.2 Expectation, Variance, and Covariance of Functions of Random Variables, 1823
	21.3 Gauss\'s Lower Bound for the Concentration of the Probability Mass in a Unimodal Distribution, 1823
	21.4 Gauss\'s Theory of Linear Minimum Variance Estimation, 1821 and 1823
	21.5 The Theorem on the Linear Unbiased Minimum Variance Estimate, 1823
	21.6 The Best Estimate of a Linear Function of the Parameters, 1823
	21.7 The Unbiased Estimate of σ² and Its Variance, 1823
	21.8 Recursive Updating of the Estimates by an Addmonal Observation, 1823
	21.9 Estimation under Linear Constraints, 1828
	21.10 A Review
PART IV. Selected Topics in Estimation Theory 1830-1930
	22. On Error and Estimation Theory, 1830-1890
		22.1 Bibliographies on the Method of Least Squares
		22.2 State of Estimation Theory around 1830
		22.3 Discussions on the Method of Least Squares and Some Alternatives
	23. Bienaymé\'s Proof of the Multivariate Central Limit Theorem and His Defense of Laplace\'s Theory of Linear Estimation, 1852 and 1853
		23.1 The Multivariate Central Limit Theorem, 1852
		23.2 Bravais\'s Confidence Ellipsoids, 1846
		23.3 Bienaymé\'s Confidence Ellipsoids and the χ² Distribution, 1852
		23.4 Bienaymé\'s Criticism of Gauss, 1853
		23.5 The Bienaymé Inequality, 1853
	24. Cauchy\'s Method for Determining the Number of Terms To Be Included in the Linear Model and for Estimating the Parameters, 1835-1853
		24.1 The Problem
		24.2 Solving the Problem by Means of the Instrumental Variable ±1, 1835
		24.3 Cauchy\'s Two-factor Multiplicative Model, 1835
		24.4 The Cauchy-Bienaymé Dispure on the Validity of the Method of Least Squares, 1853
	25. Orthogonalization and Polynomial Regression
		25.1 Orthogonal Polynomials Derived by Laplacean Orthogonalization
		25.2 Chebyshev\'s Orthogonal Polynomials, Least Squares, and Continued Fractions, 1855 and 1859
		25.3 Chebyshev\'s Orthogonal Polynomials for Equidistant Arguments, 1864 and 1875
		25.4 Gram\'s Derivation of Orthogonal Functions by the Method of Least Squares, 1879, 1883, and 1915
		25.5 Thiele\'s Free Functions and His Orthogonalization of the Linear Model, 1889, 1897, and 1903
		25.6 Schmidt\'s Orthogonalization Process, 1907 and 1908
		25.7 Notes on the Literature After 1920 on Least Squares Approximation by Orthogonal Polynomials with Equidistant Arguments
	26. Statistical Laws in the Social and Biological Sciences. Poisson, Quetelet, and Galton, 1830-1890
		26.1 Probability Theory in the Social Sciences by Condorcet and Laplace
		26.2 Poisson, Bienaymé, and Cournot on the Law of Large Numbers and Its Applications, 1830-1843
		26.3 Quetelet on the Average Man, 1835, and on the Variation around the Average, 1846
		26.4 Galton on Heredity, Regression, and Correlation, 1869-1890
		26.5 Notes on the Early History of Regression and Correlation, 1889-1907
	27. Sampling Distributions under Normality
		27.1 The Helmert Distribution, 1876, and Its Generalization to the Linear Model by Fisher, 1922
		27.2 The Distribution of the Mean Deviation by Helmert, 1876, and by Fisher, 1920
		27.3 Thiele\'s Method of Estimation and the Canonical Form of the Linear Normal Model, 1889 and 1903
		27.4 Karl Pearson\'s CHI-squared Test of Goodness of Fit, 1900, and Fisher\'s Amendment, 1924
		27.5 \"Student\'s\" t Distribution by Gosset, 1908
		27.6 Studentization, the F Distribution, and the Analysis of Variance by Fisher, 1922-1925
		27.7 The Distribution of the Correlation Coefficient, 1915, the Partial Correlation Coefficient, 1924, the Multiple Correlation Coefficient, 1928, and the Noncentral χ² and F Distributions, 1928, by Fisher
	28. Fisher\'s Theory of Estimation, 1912-1935, and His Immediate Precursors
		28.1 Notation
		28.2 On the Probable Errors of Frequency Constants by Pearson and Faon, 1898
		28.3 On the Probable Errors of Frequency Constants by Edgeworm, 1908 and 1909
		28.4 On an Absolute Criterion for Fitting Frequency Curves by Fisher, 1912
		28.5 The Parametric Statistical Model, Sufficiency, and the Method of Maximum Likelwood. Fisher, 1922
		28.6 Efficiency and Loss of Information. Fisher, 1925
		28.7 Sufficiency, the Factorization Criterion, and the Exponential Family. Fisher, 1934
		28.8 Loss of Information by Using the Maximum Likelihood Estimate and Recovery of Information by Means of Ancillary Statistics. Fisher, 1925
		28.9 Examples of Ancillarity and Conditional Inference. Fisher, 1934 and 1935
		28.10 The Discussion of Fisher\'s 1935 Paper
		28.11 A Note on Fisher and His Books on Statistics
References
A-Bab
Bai-Ben
Ben-Ber
Ber-Bos
Bos-Cau
Cau-Che
Che-Coo
Coo-Dav
Dav-Edg
Edg-Ell
Enc-Fis
Fisher
Fisher-Fou
Fou-Gau
Gau-God
God-Hal
Hal-Her
Her-Jef
Jef-Key
Key-Lam
Lam-Lap
Laplace
Lau-Lur
Mac-Moi
Moi-New
New-Pea
Pearson
Pea-Pla
Pla-Pri
Pri-Sch
Sch-She
She-Sti
Sti-Thi
Thi-Whi
Whi-Yul
Yul-Zab
Index
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	UVWYZ