Lancaster H.O. The Chi-squared Distribution

Wiley Publications in Statistics, 1969. — 356 p.THE CHI-SQUARED DISTRIBUTION commences with a brief historical introduction to the Pearson χ2, and further historical notes, especially to Karl Pearson and R. A. Fisher, are included throughout the text. The theory and practical application of the Pearson χ2 are presented and illustrated in this book by many examples and exercises. Special attention is given to the problems of approximation to discrete distributions and the use of orthogonal functions and matrices. The book then offers a canonical description of distributions, which is fundamental to the applications of χ2 and to the computation of noncentrality parameters. A number of different proofs of the approximate distribution of the Pearson χ2 are given where the parameters are known and then a chapter is devoted to the proofs when parameters have been estimated from the data. Problems of inference as they arise in the interpretation and application of the χ2 test are discussed. The normal multivariate distributions are given and the classical Pearson estimators of correlation introduced. Contingency tables of two or more dimensions are fully treated. The monograph concludes with an extensive bibliography of over 1200 entries. The Chi-Squared Distribution provides valuable information for mathematical statisticians, biometricians, economic statisticians and psychologists.CHAPTER I. HISTORICAL SURVEY OF χ2
1. Forerunners of the Pearson χ2
2. The Contributions of K. Pearson
3. The Contributions of R. A. Fisher
4. Some Quotations of Historical InterestCHAPTER II. DISTRIBUTION THEORY
1. The Gamma Variable
2. The χ2 Variable
3. Some Properties of the χ2 Distiibution
4. Tables of the χ2 Distribution
5. The Distribution of Quadratic Forms in Normal Variables
Exercises and GamilententeCHAPTER III. DISCRETE DISTRIBUTIONS
1. Condensation and Randomized Partitions
2. Significance Tests in Discrete Distributions
3. The Normal Approximation to the Binomial Distribution
4. The Normal Approximation to the Hypergeometric and Poisson Distributions
5. The Normal or χ2 Approximation to the Multinomial Distribution
Exercises and ComplementsCHAPTER IV. ORTHOGONALITY
1. Orthogonal Matrices
2. The Formation of Orthogonal Matrices from other Orthogonal Matrices
3. Sets of Orthogonal Functions on a Finite Set of Points
4. Orthonormal Polynomials and Functions on Statistical Distributions
Exercises and ComplementsCHAPTER V. THE MULTINOMIAL DISTRIBUTION
1. Introductory
2. The Multivariate Central Limit Theorem
3. The Proofs of K. Pearson
4. Stirling’s Approximation.
5. The Proof of H. E. Soper
6. The Factorization Proof
7. The Proof by Curve Fitting (Lexis Theory).
8. Analogues of the Pearson χ2
10. Empirical Verifications of the Distribution of the Discrete χ2
11. Applications of χ2 in the Mualtmormial Distribution
Exercises and ComplementsCHAPTER VI. CANONICAL OR STANDARD FORMS FOR PROBABILITY DISTRIBUTIONS
1. Probability Measures
2. Finite Discrete Distributions in Two Dimensions
3. ϕ2-bounded Bivaniate ipicitibugions,
4. The General Bivariate Distribution.
5. Multivariate Distributions
6. Independence and Association
Exercises and ComplementsCHAPTER VII. NON-CENTRAL χ2
1. Distribution Theory
2. The Comparison of Two Normal Populations
3. Analogues of the Pearson χ2, the Combination of Probabilities
Exercises and ComplementsCHAPTER VIII. TESTS OF GOODNESS OF FIT IN THE MULTINOMIAL DISTRIBUTION
1. Introductory
2. Least Squares and Minimum χ2
3. The Fitting of Sufficient Statistics
4. χ2 in the Multinomial Distribution with Estimated Parameters (Fisher Theory)
5. Estimated Parameters (Cramér Theory)
6. Estimated Parameters and Orthonormal Theory
7. The Test of Goodness of Fit
Exercises and ComplementsCHAPTER IX. PROBLEMS OF INFERENCE
1. Introductory
2. Tests of Hypotheses
3. Significance Levels
4. The Likelihood Ratio Test and χ2
5. Multiple Comparisons
6. Grouping, or Choice of Partitions of the Measure Space
7. Large Values of χ2
8. Small Values of χ2
9. Hidden Parameters
10. χ2 and the Sample Size
11. Small Class Frequencies
12. The Partition of χ2
13. Misclassification and Missing Values
14. The Reconciliation of χ2
15. Miscellaneous Inference
Exercises and ComplementsCHAPTER X. NORMAL CORRELATION
1. Introductory
2. The Partial Correlations
3. The Canonical Correlations of Hotelling
4. Kolmogorov’s Canonical Problem
5. Multivariate Normality
6. The Canonical Correlations
7. The Wishart Distribution
8. Tetrachoric Correlation
9. The Polychoric Series
10. The Correlation Ratio
11. Biserial η
12. Tests of Normality
13. Various Measures of Correlation
Exercises and ComplementsCHAPTER XI. TWO-WAY CONTINGENCY TABLES
1. Introductory
2. Probability Models in a Two-way Contingency Table
3. Tests of Independence
4. The Fourfold Table
5. Combinatorial Theory of the Two-way Tables
6. Asymptotic Theory of the Two-way Tables
7. Parameters of Non-Centrality
8. Symmetry and sei aa in Two-way Tables
9. Reparametrization
10. Measures of Association
11. The Homogeneity of Several Populations
12. Contingency Tables — Miscellaneous Topics
Exercises and ComplementsXII. CONTINGENCY TABLES OF HIGHER DIMENSIONS
1. Introductory and Historical
2. Interactions and Generalized Correlations
3. Models
4. Combinatorial Theory
5. The Fisher-Bartlett Methods
6. Asymptotic Theory
7. Canonical Variables
8. Exchangeable Random Variables and Symmetry
Exercises and Complements