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