# MATHEMATICAL METHODS OF STATISTICS

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1 MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946

2 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS 1 3. SETS OF POINTS Page Chapter 1. General properties of sets 3 1. Sets. 2. Subsets, space. 3. Operations on sets. 4. Sequences of sets. 5. Monotone sequences. 6. Additive classes of sets. Chapter 2. Linear point sets ' Intervals. 2. Various properties of sets in R Borel sets. Chapter 3. Point sets in n dimensions Intervals. 2. Various properties of sets in Rn- 3. Borel sets. 4. Linear sets. 5. Subspace, product space. References to chapters CHAPTERS 4 7. THEORY OF MEASURE AND INTEGRATION Chapter 4. The Lebesgue measure of a linear point set 19 v 1. Length of an interval. < 2. Generalization. Z. The measure of a sum of intervals. *"4. Outer and inner measure of a bounded set. 5. Measurable -sets and Lebesgue measure. 6.' The class of measurable sets. ' ' 1. Measurable sets and Borel sets. Chapter 5. The Lebesgue integral for functions of one variable The integral of a bounded function over a set of finite measure. "2. immeasurable functions. *3. Properties of the integral. ; 4. The integral of an unbounded function over a set of finite measure. -6. The integral over a set of infinite measure. 6. The Lebesgue integral-as an additive set function.. Chapter 6. Non-negative additive set functions in R l '.' Generalization of the Lebesgue measure and the Lebesgue integral. 2. Set functions and point functions. 3. Construction of a set function. 4. Pmea'sure. 6. Bounded set functions. 6. Distributions. 7. Sequences of distributions. 8. A convergence theorem. XI

3 Page Chapter 7. The Lebesgue-Stieltjes integral for functions of one variable The integral of a bounded function over a set of finite P-measure. 2. Unbounded functions and sets of infinite P-measure. 3. Lebesgue-Stieltjes integrals with a parameter. 4. Lebesgue-Stieltjes integrals with respect to a distribution. 6. The Riemann-Stieltjes integral. References to chapters CHAPTERS 8 9. THEORY OF MEASURE AND INTEGRATION Chapter 8. Lebesgue measure and other additive set functions inr, Lebesgue measure in Rn. 2. Non-negative additive set functions in Rn. 3. Bounded set functions. 4. Distributions. 5. Sequences of distributions. 6. Distributions in a product space. Chapter 9. The Lebesgue-Stieltjes integral for functions of n variables The Lebesgue-Stieltjes integral. 2. Lebesgue-Stieltjes integrals with respect to a distribution. 3. A theorem on repeated integrals. 4. The Riemann-Stieltjes integral. 6. The Schwarz inequality. CHAPTERS VARIOUS QUESTIONS. Chapter 10. Fourier integrals The characteristic function of a distribution in R l. 2. Some auxiliary functions. 3. Uniqueness theorem for characteristic functions in R t. 4. Continuity theorem for characteristic functions in JR,. 5. Some particular integrals. 6. The characteristic function of a distribution in Rn. 7. Continuity theorem for characteristic functions in Rn. Chapter 11. Matrices, determinants and quadratic forms Matrices. 2. Vectors. 3. Matrix notation for linear transformations. 4. Matrix notation for bilinear and quadratic forms. 5. Determinants. 6. Rank. 7. Adjugate and reciprocal matrices. 8. Linear equations. 9. Orthogonal matrices. Characteristic numbers. 10. Nonnegative quadratic forms. 11. Decomposition of S a;*. 12. Some integral formulae. Chapter 12. Miscellaneous complements The symbols 0, o and co. 2. The Euler-MacLaurin sum formula. 3. The Gamma function. 4. The Beta function. 5. Stirling's formula. 6. Orthogonal polynomials. XII

4 Second Part. RANDOM VARIABLES AND PROBABILITY DISTRIBU- TIONS. CHAPTERS FOUNDATIONS. Page Chapter 13. Statistics and probability Random experiments. 2. Examples. 3. Statistical regularity. 4. Object of a mathematical theory. 5. Mathematical probability. Chapter 14. Fundamental definitions and axioms Random variables. (Axioms 1 2.) 2. Combined variables. (Axiom 3.) 3. Conditional distributions. 4. Independent variables. 6. Functions of random variables. 6. Conclusion. CHAPTERS VARIABLES AND DISTRIBUTIONS IN i? x. Chapter 15. General properties Distribution function and frequency function. 2. Two simple types of distributions. 3. Mean values. 4. Moments. 5. Measures of location. 6. Measures of dispersion. 7. Tchebycheff's theorem. 8. Measures of skewness and excess. 9. Characteristic functions. 10. Semi-invariants. 11. Independent variables. 12. Addition'of independent variables. Chapter 16. Various discrete distributions The function e (x). 2. The binomial distribution. 3. Bernoulli's theorem. 4. De Moivre's theorem. 5. The Poisson distribution. 6. The generalized binomial distribution of Poisson. Chapter 17. The normal distribution The normal functions. 2. The normal distribution. 3. Addition of independent normal variables. 4. The central limit theorem. 5. Complementary remarks to the central limit theorem. 6. Orthogonal expansion derived from the normal distribution Asymptotic expansion derived from the normal distribution. 8. The role of the normal distribution in statistics. '-. Chapter 18. Various distributions related to the normal The x'distribution. 2. Stu'dent's distribution. 3. Fisher's zdistribution. 4. The Beta distribution. Chapter 19. Further continuous distributions The rectangular distribution. 2. Cauchy's and Laplace's distributions. 3. Truncated distributions. 4. The Pearson system. XIII

5 Page Chapter 20. Some convergence theorems Convergence of distributions and variables. 2. Convergence of certain distributions to the normal. 3. Convergence in probability., 4. Tchebyeheff's theorem. 6. Khintohine's theorem. 6. A convergence theorem. Exercises to chapters CHAPTERS VARIABLES AND DISTRIBUTIONS IN JR n. Chapter 21. The two-dimensional case ' Two simple types of distributions. 2. Mean values, moments. 3. Characteristic functions. 4. Conditional distributions. 6. Regression, I. 6. Regression, II. 7. The correlation 'coefficient. 8. Linear- transformation of variables The correlation ratio and the mean square contingency. 10. The ellipse of concentration. 11. Addition of independent variables. 12. The normal distribution. Chapter 22. General properties of distributions in R n Two simple types of distributions. Conditional distributions. 2. Chauge of variables in a continuous distribution. 3. Mean values, moments. 4. Characteristic functions. 5. Rank of a distribution. 6. Linear transformation of variables. 7. The ellipsoid of concentration. Chapter 23. Regression and correlation in n variables Regression surfaces. 2. Linear mean square regression. 3. Residuals. 4. Partial correlation. 5. The multiple correlation coefficient. 6. Orthogonal mean square regression. ' Chapter 24. The normal distribution The characteristic function. 2. The non-singular normal distribution. 3. The singular normal distribution. 4. Linear transformation of normally distributed variables. 6. Distribution of a sum of squares. 6. Conditional distributions. 7. Addition of independent variables. The central limit theorem. Exercises to chapters " Third Part, STATISTICAL INFERENCE. CHAPTERS GENERALITIES. Chapter 25. Preliminary notions on sampling Introductory remarks. 2: Simple random sampling. 3. The distribution of the sample. 4. The sample values as random variables. Sampling XIV

6 distributions. ' 5. Statistical image of a distribution. 6. Biased sampling. Random sampling numbers. 7. Sampling without replacement. The representative method. Chapter 26. Statistical inference.., Introdnctory remarks. 2. Agreement between theory and facts. Tests of significance. 3. Description. 4, Analysis. 5. Prediction. CHAPTERS SAMPLING DISTRIBUTIONS. Chapter 27. Characteristics of sampling distributions Notations. 2. The sample mean x. 3. The moments a v. 4. The variance m t. 5. Higher central moments and semi-invariants. 6. Unbiased estimates. 7. Functions of moments. 8. Characteristics of multidimensional distributions. 9. Corrections for grouping. Chapter 28. Asymptotic properties of sampling distributions Introductory remarks. 2. The moments. 3. The central moments. 4. Functions of moments. 5. The quantiles. 6. The extreme values and the range. Chapter 29. Exact sampling distributions The problem. 2. Fisher's lemma. Degrees of freedom. 3. Trie joint distribution of x and s* in samples from a normal distribution. 4. Student's ratio. 5. A lemma. 6. Sampling from a two-dimensional normal distribution. 7. The correlation coefficient. 8. The regression coefficients. 9. Sampling from a fc-dimensional normal distribution. 10. The generalized variance. 11. The generalized Student ratio. 12. Regression coefficients. 13. Partial and multiple correlation coefficients. CHAPTERS TESTS OF SIGNIFICANCE, I. Chapter 30. Tests of goodness of fit and allied tests The x 2 test in the case of a completely specified hypothetical distribution. 2. Examples. 3. The y* test when certain parameters are estimated - from the sample Examples. 5. Contingency tables. 6. j; 1 as a test of homogeneity. 7. Criterion of differential death rates. 8. Further tests of goodness of fit. Chapter 31. Tests of significance for parameters Tests based on standard errors. 2. Tests based on exact distributions. 3. Examples. CHAPTERS THEORY OF ESTIMATION. Chapter 32. Classification of estimates The problem. 2. Two-lemmas. 3. Minimum variance of an estimate. XV

7 Efficient estimates. 4. Sufficient estimates. 5. Asymptotically efficient estimates. 6. The case of two unknown parameters. 7. Several unknown parameters. 8. Generalization. Page Chapter 33. Methods of estimation The method of moments. 2. The method of maximum likelihood. 3. Asymptotic properties of maximum likelihood estimates. 4. The JJ* minimum method. Chapter 34. Confidence regions Introductory remarks. 2. A single unknown parameter. 3. The general case. 4. Examples. CHAPTERS TESTS OF SIGNIFICANCE, II. Chapter 35. General theory of testing statistical hypotheses The choice of a test of significance. 2. Simple and composite hypotheses. 3. Tests of simple hypotheses. Most powerful tests. 4. Unbiased tests. 5. Tests of composite hypotheses. Chapter 36. Analysis of variance Variability of mean values. 2. Simple grouping of variables. 3. Generalization. 4. Randomized blocks. 6. Latin squares. Chapter 37. Some regression problems Problems involving non-random variables. 2. Simple regression. 3. Multiple regression. 4. Further regression problems. TABLES 1 2. THE NORMAL DISTRIBUTION 557 TABLE 3. THE ^-DISTRIBUTION 559 TABLE 4. THE ^-DISTRIBUTION 560 LIST OF REFERENCES 561 INDEX XVI

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