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 are 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.I. HISTORICAL SURVEY OF χ2.
Forerunners of the Pearson χ2.
The Contributions of K. Pearson.
The Contributions of R. A. Fisher.
Some Quotations of Historical Interest.II. DISTRIBUTION THEORY.
The Gamma Variable.
The χ2 Variable.
Some Properties of the χ2 Distribution.
Tables of the χ2 Distribution.
The Distribution of Quadratic Forms in Normal Variables.
Exercises and Gamilentente.III. DISCRETE DISTRIBUTIONS.
Condensation and Randomized Partitions.
Significance Tests in Discrete Distributions.
The Normal Approximation to the Binomial Distribution.
The Normal Approximation to the Hypergeometric and Poisson Distributions.
The Normal or χ2 Approximation to the Multinomial Distribution.
Exercises and Complements.IV. ORTHOGONALITY.
Orthogonal Matrices.
The Formation of Orthogonal Matrices from other Orthogonal Matrices.
Sets of Orthogonal Functions on a Finite Set of Points.
Orthonormal Polynomials and Functions on Statistical Distributions.
Exercises and Complements.
V. THE MULTINOMIAL DISTRIBUTION.
Introductory.
The Multivariate Central Limit Theorem.
The Proofs of K. Pearson.
Stirling’s Approximation.
The Proof of H. E. Soper.
The Factorization Proof.
The Proof by Curve Fitting (Lexis Theory).
Analogs of the Pearson χ2.
Empirical Verifications of the Distribution of the Discrete χ2.
Applications of χ2 in the Mualtmormial Distribution.
Exercises and Complements.VI. CANONICAL OR STANDARD FORMS FOR PROBABILITY DISTRIBUTIONS.
Probability Measures.
Finite Discrete Distributions in Two Dimensions.
ϕ2-bounded Bivaniate ipicitibugions,
The General Bivariate Distribution.
Multivariate Distributions.
Independence and Association.
Exercises and Complements.
VII. NON-CENTRAL χ2.
Distribution Theory.
The Comparison of Two Normal Populations.
Analogs of the Pearson χ2, the Combination of Probabilities.
Exercises and Complements.VIII. TESTS OF GOODNESS OF FIT IN THE MULTINOMIAL DISTRIBUTION.
Introductory.
Least Squares and Minimum χ2.
The Fitting of Sufficient Statistics.
χ2 in the Multinomial Distribution with Estimated Parameters (Fisher Theory).
Estimated Parameters (Cramér Theory).
Estimated Parameters and Orthonormal Theory.
The Test of Goodness of Fit.
Exercises and Complements.IX. PROBLEMS OF INFERENCE.
Introductory.
Tests of Hypotheses.
Significance Levels.
The Likelihood Ratio Test and χ2.
Multiple Comparisons.
Grouping, or Choice of Partitions of the Measure Space.
Large Values of χ2.
Small Values of χ2.
Hidden Parameters.
χ2 and the Sample Size.
Small Class Frequencies.
The Partition of χ2.
Misclassification and Missing Values.
The Reconciliation of χ2.
Miscellaneous Inference.
Exercises and Complements.X. NORMAL CORRELATION.
Introductory.
The Partial Correlations.
The Canonical Correlations of Hotelling.
Kolmogorov’s Canonical Problem.
Multivariate Normality.
The Canonical Correlations.
The Wishart Distribution.
Tetrachoric Correlation.
The Polychoric Series.
The Correlation Ratio.
Biserial η.
Tests of Normality.
Various Measures of Correlation.
Exercises and Complements.XI. TWO-WAY CONTINGENCY TABLES.
Introductory.
Probability Models in a Two-way Contingency Table.
Tests of Independence.
The Fourfold Table.
Combinatorial Theory of the Two-way Tables.
Asymptotic Theory of the Two-way Tables.
Parameters of Non-Centrality.
Symmetry and sei aa in Two-way Tables.
Reparametrization.
Measures of Association.
The Homogeneity of Several Populations.
Contingency Tables — Miscellaneous Topics.
Exercises and Complements.XII. CONTINGENCY TABLES OF HIGHER DIMENSIONS.
Introductory and Historical.
Interactions and Generalized Correlations.
Models.
Combinatorial Theory.
The Fisher-Bartlett Methods.
Asymptotic Theory.
Canonical Variables.
Exchangeable Random Variables and Symmetry.
Exercises and Complements.