| Algebra of Vectors and Matrices / Chapter 1： |
| Vector Spaces |
| Definition of Vector Spaces and Subspaces / 1a.1： |
| Basis of a Vector Space / 1a.2： |
| Linear Equations / 1a.3： |
| Vector Spaces with an Inner Product / 1a.4： |
| Complements and Problems |
| Theory of Matrices and Determinants / lb： |
| Matrix Operations / 1b.1： |
| Elementary Matrices and Diagonal Reduction of a Matrix / 1b.2： |
| Determinants / 1b.3： |
| Transformations / 1b.4： |
| Generalized Inverse of a Matrix / 1b.5： |
| Matrix Representation, of Vector Spaces, Bases, etc / 1b.6： |
| Idempotent Matrices / 1b.7： |
| Special Products of Matrices / 1b.8： |
| Eigenvalues and Reduction of Matrices / 1c： |
| Classification and Transformation of Quadratic Forms / 1c.1： |
| Roots of Determinantal Equations / 1c.2： |
| Canonical Reduction of Matrices / 1c.3： |
| Projection Operator / 1c.4： |
| Further Results on g-Inverse / 1c.5： |
| Restricted Eigenvalue Problem / 1c.6： |
| Convex Sets in Vector Spaces / 1d： |
| Definitions / 1d.1： |
| Separation Theorems for Convex Sets / 1d.2： |
| Inequalities / 1e： |
| Cauchy-Schwarz (C-S) Inequality / 1e.1： |
| Holder?s Inequality / 1e.2： |
| Hadamard?s Inequality / 1e.3： |
| Inequalities Involving Moments / 1e.4： |
| Convex Functions and Jensen?s Inequality / 1e.5： |
| Inequalities in Information Theory / 1e.6： |
| Stirling?s Approximation / 1e.7： |
| Extrema of Quadratic Forms / 1f： |
| General Results / 1f.1： |
| Results Involving Eigenvalues and Vectors / 1f.2： |
| Minimum Trace Problems / 1f.3： |
| Probability Theory, Tools and Techniques / Chapter 2： |
| Calculus of Probability / 2a： |
| The Space of Elementary Events / 2a.l： |
| The Class of Subsets (Events) / 2a.2： |
| Probability as a Set Function / 2a.3： |
| Borel Field (&sigma-field) and Extension of Probability Measure / 2a.4： |
| Notion of a Random Variable and Distribution Function / 2a.5： |
| Multidimensional Random Variable / 2a.6： |
| Conditional Probability and Statistical Independence / 2a.7： |
| Conditional Distribution of a Random Variable / 2a.8： |
| Mathematical Expectation and Moments of Random Variables / 2b： |
| Properties of Mathematical Expectation / 2b.1： |
| Moments, 2b.3 Conditional Expectation / 2b.2： |
| Characteristic Function (c.f.) / 2b.4： |
| Inversion Theorems / 2b.5： |
| Multivariate Moments / 2b.6： |
| Limit Theorems / 2c： |
| Kolmogorov Consistency Theorem / 2c.1： |
| Convergence of a Sequence of Random Variables / 2c.2： |
| Law of Large Numbers / 2c.3： |
| Convergence of a Sequence of Distribution Functions / 2c.4： |
| Central Limit Theorems / 2c.5： |
| Sums of Independent Random Variables / 2c.6： |
| Family of Probability Measures and Problems of Statistics / 2d： |
| Family of Probability Measures / 2d.1： |
| The Concept of a Sufficient Statistic / 2d.2： |
| Characterization of Sufficiency / 2d.3： |
| Stieltjes and Lebesgue Integrals / Appendix 2A： |
| Some Important Theorems in Measure Theory and Integration / Appendix 2B： |
| Invariance / Appendix 2C： |
| Statistics, Subfields, and Sufficiency / Appendix 2D： |
| Non-Negative Definiteness of a Characteristic Function / Appendix 2E： |
| Complements and Problems Chapter 3: Continuous Probability Models |
| Univariate Models / 3a： |
| Normal Distribution / 3a.1： |
| Gamma Distribution / 3a.2： |
| Beta Distribution / 3a.3： |
| Cauchy Distribution / 3a.4： |
| Student?s t Distribution / 3a.5： |
| Distributions Describing Equilibrium States in Statistical Mechanics / 3a.6： |
| Distribution on a Circle / 3a.7： |
| Sampling Distributions / 3b： |
| Definitions and Results / 3b.1： |
| Sum of Squares of Normal Variables / 3b.2： |
| Joint Distribution of the Sample Mean and Variance / 3b.3： |
| Distribution of Quadratic Forms / 3b.4： |
| Three Fundamental Theorems of the least Squares Theory / 3b.5： |
| The p-Variate Normal Distribution / 3b.6： |
| The Exponential Family of Distributions / 3b.7： |
| Symmetric Normal Distribution / 3c： |
| Definition / 3c.1： |
| Bivariate Normal Distribution / 3c.2： |
| General Properties / 3d.1： |
| The Theory of least Squares and Analysis of Variance / 3d.2： |
| Theory of least Squares / Linear Estimation)4a： |
| Gauss-Markoff Setup (Y, Xβ, σ2I) / 4a.1： |
| Normal Equations and least Squares (l.s.) Estimators / 4a.2： |
| g-Inverse and a Solution of the Normal Equation / 4a.3： |
| Variances and Covariances of l.s. Estimators / 4a.4： |
| Estimation of σ2 / 4a.5： |
| Other Approaches to the l.s. Theory / Geometric Solution)4a.6： |
| Explicit Expressions for Correlated Observations / 4a.7： |
| Some Computational Aspects of the l.s. Theory / 4a.8： |
| least Squares Estimation with Restrictions on Parameters / 4a.9： |
| Simultaneous Estimation of Parametric Functions / 4a.10： |
| least Squares Theory when the Parameters Are Random Variables / 4a.11： |
| Choice of the Design Matrix / 4a.12： |
| Tests of Hypotheses and Interval Estimation / 4b： |
| Single Parametric Function (Inference) / 4b.1： |
| More than One Parametric Function (Inference) / 4b.2： |
| Setup with Restrictions / 4b.3： |
| Problems of a Single Sample / 4c： |
| The Test Criterion / 4c.1： |
| Asymmetry of Right and left Femora (Paired Comparison) / 4c.2： |
| One-Way Classified Data / 4d： |
| An Example / 4d.1： |
| Two-Way Classified Data / 4e： |
| Single Observation in Each Cell / 4e.1： |
| Multiple but Equal Numbers in Each Cell / 4e.2： |
| Unequal Numbers in Cells / 4e.3： |
| A General Model for Two-Way Data and Variance Components / 4f： |
| A General Model / 4f.1： |
| Variance Components Model / 4f.2： |
| Treatment of the General Model / 4f.3： |
| The Theory and Application of Statistical Regression / 4g： |
| Concept of Regression (General Theory) / 4g.1： |
| Measurement of Additional Association / 4g.2： |
| Prediction of Cranial Capacity (a Practical Example) / 4g.3： |
| Test for Equality of the Regression Equations / 4g.4： |
| The Test for an Assigned Regression Function / 4g.5： |
| Restricted Regression / 4g.6： |
| The General Problem of least Squares with Two Sets of Parameters / 4h： |
| Concomitant Variables / 4h.1： |
| Analysis of Covariance / 4h.2： |
| An Illustrative Example / 4h.3： |
| Unified Theory of Linear Estimation / 4i： |
| A Basic Lemma on Generalized Inverse / 4i.1： |
| The General Gauss-Markoff Model (GGM) / 4i.2： |
| The Inverse Partitioned Matrix (IPM) Method / 4i.3： |
| Untried Theory of Least Squares / 4i.4： |
| Estimation of Variance Components / 4j： |
| Minque Theory / 4j.1： |
| Computation under the Euclidian Norm / 4j.3： |
| Biased Estimation in Linear Models / 4k： |
| Best Linear Estimator (BLE) / 4k.1： |
| Best Linear Minimum Bias Estimation (BLIMBE) Complements and Problems / 4k.2： |
| Criteria and Methods of Estimation / Chapter 5： |
| Minimum Variance Unbiased Estimation / 5a： |
| Minimum Variance Criterion / 5a.1： |
| Some Fundamental Results on Minimum Variance Estimation / 5a.2： |
| The Case of Several Parameters / 5a.3： |
| Fisher?s Information Measure / 5a.4： |
| An Improvement of Un-biased Estimators / 5a.5： |
| General Procedures / 5b： |
| Statement of the General Problem (Bayes Theorem) / 5b.1： |
| Joint d.f. of (&Teata;, x) Completely Known / 5b.2： |
| The Law of Equal Ignorance / 5b.3： |
| Empirical Bayes Estimation Procedures / 5b.4： |
| Fiducial Probability / 5b.5： |
| Minimax Principle / 5b.6： |
| Principle of Invariance / 5b.7： |
| Criteria of Estimation in Large Samples / 5c： |
| Consistency / 5c.1： |
| Efficiency / 5c.2： |
| Some Methods of Estimation in Large Samples / 5d： |
| Method of Moments / 5d.1： |
| Minimum Chi-Square and Associated Methods / 5d.2： |
| Maximum Likelihood / 5d.3： |
| Estimation of the Multinomial Distribution / 5e： |
| Nonparametric Case / 5e.1： |
| Parametric Case / 5e.2： |
| Estimation of Parameters in the General Case / 5f： |
| Assumptions and Notations / 5f.1： |
| Properties of m.l. Equation Estimators / 5f.2： |
| The Method of Scoring for the Estimation of Parameters, Complements and Problems / 5g： |
| Large Sample Theory and Methods / Chapter 6： |
| Some Basic Results / 6a： |
| Asymptotic Distribution of Quadratic Functions of Frequencies / 6a.1： |
| Some Convergence Theorems / 6a.2： |
| Chi-Square Tests for the Multinomial Distribution / 6b： |
| Test of Departure from a Simple Hypothesis / 6b.1： |
| Chi-Square Test for Goodness of Fit / 6b.2： |
| Test for Deviation in a Single Cell / 6b.3： |
| Test Whether the Parameters Lie in a Subset / 6b.4： |
| Some Examples / 6b.5： |
| Test for Deviations in a Number of Cells / 6b.6： |
| Tests Relating to Independent Samples from Multinomial Distributions / 6c： |
| Test of Homogeneity of Parallel Samples / 6c.1： |
| Contingency Tables / 6c.3： |
| The Probability of an Observed Configuration and Tests in Large Samples / 6d.1： |
| Tests of Independence in a Contingency Table / 6d.2： |
| Tests of Independence in Small Samples / 6d.3： |
| Some General Classes of Large Sample Tests / 6e： |
| Notations and Basic Results / 6e.1： |
| Test of a Simple Hypothesis / 6e.2： |
| Test of a Composite Hypothesis / 6e.3： |
| Order Statistics / 6f： |
| The Empirical Distribution Function / 6f.1： |
| Asymptotic Distribution of Sample Fractiles / 6f.2： |
| Transformation of Statistics / 6g： |
| A General Formula / 6g.1： |
| Square Root Transformation of the Poisson Variate / 6g.2： |
| Sin-1 Transformation of the Square Root of the Binomial Proportion / 6g.3： |
| Tanh-1 Transformation of the Correlation Coefficient / 6g.4： |
| Standard Errors of Moments and Related Statistics / 6h： |
| Variances and Covariances of Raw Moments / 6h.1： |
| Asymptotic Variances and Covariances of Central Moments / 6h.2： |
| Exact Expressions for Variances and Covariances of Central Moments / 6h.3： |
| Theory of Statistical Inference / Chapter 7： |
| Testing of Statistical Hypotheses / 7a： |
| Statement of the Problem / 7a.1： |
| Neyman-Pearson Fundamental Lemma and Generalizations / 7a.2： |
| Simple Ho against Simple H / 7a.3： |
| Locally Most Powerful Tests / 7a.4： |
| Testing a Composite Hypothesis / 7a.5： |
| Fisher-Behrens Problem / 7a.6： |
| Asymptotic Efficiency of Tests / 7a.7： |
| Confidence Intervals / 7b： |
| The General Problem / 7b.1： |
| A General Method of Constructing a Confidence Set / 7b.2： |
| Set Estimators for Functions of &Teata; / 7b.3： |
| Sequential Analysis / 7c： |
| Wald?s Sequential Probability Ratio Test / 7c.1： |
| Some Properties of the S.P.R.T / 7c.2： |
| Efficiency of the S.P.R.T / 7c.3： |
| An Example of Economy of Sequential Testing / 7c.4： |
| The Fundamental Identity of Sequential Analysis / 7c.5： |
| Sequential Estimation / 7c.6： |
| Sequential Tests with Power One / 7c.7： |
| Problem of Identification?Decision Theory / 7d： |
| Randomized and Nonrandomized Decision Rules / 7d.1： |
| Bayes Solution / 7d.3： |
| Complete Class of Decision Rules / 7d.4： |
| Minimax Rule / 7d.5： |
| Nonparametric Inference / 7e： |
| Concept of Robustness / 7e.1： |
| Distribution-Free Methods / 7e.2： |
| Some Nonparametric Tests / 7e.3： |
| Principle of Randomization / 7e.4： |
| Ancillary Information / 7f： |
| Multivariate Analysis / Chapter 8： |
| Multivariate Normal Distribution / 8a： |
| Properties of the Distribution / 8a.1： |
| Some Characterizations of Np / 8a.3： |
| Density Function of the Multivariate Normal Distribution / 8a.4： |
| Estimation of Parameters / 8a.5： |
| Np as a Distribution with Maximum Entropy / 8a.6： |
| Wishart Distribution / 8b： |
| Definition and Notation / 8b.1： |
| Some Results on Wishart Distribution / 8b.2： |
| Analysis of Dispersion / 8c： |
| The Gauss-Markoff Setup for Multiple Measurements / 8c.1： |
| Tests of Linear Hypotheses, Analysis of Dispersion (A.D.) / 8c.2： |
| Test for Additional Information / 8c.4： |
| The Distribution of A / 8c.5： |
| Test for Dimensionality / Structural Relationship)8c.6： |
| Analysis of Dispersion with Structural Parameters (Growth Model) / 8c.7： |
| Some Applications of Multivariate Tests / 8d： |
| Test for Assigned Mean Values / 8d.1： |
| Test for a Given Structure of Mean Values / 8d.2： |
| Test for Differences between Mean Values of Two Populations / 8d.3： |
| Test for Differences in Mean Values between Several Populations / 8d.4： |
| Barnard?s Problem of Secular Variations in Skull Characters / 8d.5： |
| Discriminatory Analysis (Identification) / 8e： |
| Discriminant Scores for Decision / 8e.1： |
| Discriminant Analysis in Research Work / 8e.2： |
| Discrimination between Composite Hypotheses / 8e.3： |
| Relation between Sets of Variates / 8f： |
| Canonical Correlations / 8f.1： |
| Properties of Canonical Variables / 8f.2： |
| Effective Number of Common Factors / 8f.3： |
| Factor Analysis / 8f.4： |
| Orthonormal Basis of a Random Variable / 8g： |
| The Gram-Schmidt Basis / 8g.1： |
| Principal Component Analysis / 8g.2： |
| Publications of the Author |
| Author Index |
| Subject Index |
| Continuous Probability Models |
| The Theory of Least Squares and Analysis of Variance |
ＥＢ
