Algebra of Vectors and Matrices |
Probability Theory, Tools and Techniques |
Continuous Probability Models |
The Theory of Least Squares and Analysis of Variance |
Criteria and Methods of Estimation |
Large Sample Theory and Methods |
Theory of Statistical Inference |
Multivariate Analysis |
Publications of the Author |
Author Index |
Subject Index |
Vector Spaces / Chapter 1: |
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: |
Calculus of Probability / Chapter 2: |
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: |
Minimum Variance Unbiased Estimation / Chapter 5: |
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: |
Some Basic Results / Chapter 6: |
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: |
Testing of Statistical Hypotheses / Chapter 7: |
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 Normal Distribution / Chapter 8: |
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: |