Linear Statistical Inference and its applications

Preface......Page f007.djvu
Preface to the Second Edition......Page f011.djvu
Contents......Page f013.djvu
1 Algebra of Vectors and Matrices 1......Page p001.djvu
1a.1 Definition of Vector Spaces and Subspaces 2......Page p002.djvu
1a.2 Basis of a Vector Space 4......Page p004.djvu
1a.3 Linear Equations 6......Page p006.djvu
1a.4 Vector Spaces with an Inner Product 8......Page p008.djvu
Complements and Problems 11......Page p011.djvu
1b.1 Matrix Operations 14......Page p014.djvu
1b.2 Elementary Matrices and Diagonal Reduction of a Matrix 17......Page p017.djvu
1b.3 Determinants 22......Page p022.djvu
1b.4 Transformations 23......Page p023.djvu
1b.5 Generalized Inverse of a Matrix 24......Page p024.djvu
1b.6 Matrix Representation, of Vector Spaces, Bases, etc. 27......Page p027.djvu
1b.7 Idempotent Matrices 28......Page p028.djvu
1b.8 Special Products of Matrices 29......Page p029.djvu
Complements and Problems 30......Page p030.djvu
1c.1 Classification and Transformation of Quadratic Forms 34......Page p034.djvu
1c.2 Roots of Determinantal Equations 38......Page p038.djvu
1c.3 Canonical Reduction of Matrices 39......Page p039.djvu
1c.4 Projection Operator 46......Page p046.djvu
1c.5 Further Results on g-Inverse 48......Page p048.djvu
1c.6 Restricted Eigenvalue Problem 50......Page p050.djvu
1d.2 Separation Theorems for Convex Sets 51......Page p051.djvu
1e Inequalities 53......Page p053.djvu
1e.1 Cauchy-Schwarz (C-S) Inequality 54......Page p054.djvu
1e.2 Holder\'s Inequality 55......Page p055.djvu
1e.4 Inequalities Involving Moments 56......Page p056.djvu
1e.5 Convex Functions and Jensen\'s Inequality 57......Page p057.djvu
1e.6 Inequalities in Information Theory 58......Page p058.djvu
1e.7 Stirling\'s Approximation 59......Page p059.djvu
1f.1 General Results 60......Page p060.djvu
1f.2 Results Involving Eigenvalues and Vectors 62......Page p062.djvu
1f.3 Minimum Trace Problems 65......Page p065.djvu
Complements and Problems 67......Page p067.djvu
References 78......Page p078.djvu
2 Probability Theory, Tools and Techniques 79......Page p079.djvu
2a.1 The Space of Elementary Events 80......Page p080.djvu
2a.3 Probability as a Set Function 81......Page p081.djvu
2a.4 Borel Field (sigma-field) and Extension of Probability Measure 83......Page p083.djvu
2a.5 Notion of a Random Variable and Distribution Function 84......Page p084.djvu
2a.6 Multidimensional Random Variable 89......Page p089.djvu
2a.7 Conditional Probability and Statistical Independence 90......Page p090.djvu
2a.8 Conditional Distribution of a Random Variable 91......Page p091.djvu
2b.1 Properties of Mathematical Expectation 92......Page p092.djvu
2b.2 Moments 93......Page p093.djvu
2b.3 Conditional Expectation 96......Page p096.djvu
2b.4 Characteristic Function (c.f.) 99......Page p099.djvu
2b.5 Inversion Theorems 104......Page p104.djvu
2b.6 Multivariate Moments 107......Page p107.djvu
2c.1 Kolmogorov Consistency Theorem 108......Page p108.djvu
2c.2 Convergence of a Sequence of Random Variables 110......Page p110.djvu
2c.3 Law of Large Numbers 111......Page p111.djvu
2c.4 Convergence of a Sequence of Distribution Functions 116......Page p116.djvu
2c.5 Central Limit Theorems 126......Page p126.djvu
2c.6 Sums of Independent Random Variables 129......Page p129.djvu
2d.2 The Concept of a Sufficient Statistic 130......Page p130.djvu
2d.3 Characterization of Sufficiency 131......Page p131.djvu
2A Stieltjes and Lebesgue Integrals 132......Page p132.djvu
2B Some Important Theorems in Measure Theory and Integration 134......Page p134.djvu
2C Invariance 138......Page p138.djvu
2D Statistics, Subfields, and Sufficiency 139......Page p139.djvu
2E Non-Negative Definiteness of a Characteristic Function 141......Page p141.djvu
Complements and Problems 142......Page p142.djvu
References 153......Page p153.djvu
3 Continuous Probability Models 155......Page p155.djvu
3a.1 Normal Distribution 158......Page p158.djvu
3a.2 Gamma Distribution 164......Page p164.djvu
3a.3 Beta Distribution 168......Page p168.djvu
3a.4 Cauchy Distribution 169......Page p169.djvu
3a.5 Student\'s t Distribution 170......Page p170.djvu
3a.6 Distributions Describing Equilibrium States in Statistical Mechanics 172......Page p172.djvu
3a.7 Distribution on a Circle 175......Page p175.djvu
3b.1 Definitions and Results 179......Page p179.djvu
3b.2 Sum of Squares of Normal Variables 181......Page p181.djvu
3b.3 Joint Distribution of the Sample Mean and Variance 182......Page p182.djvu
3b.4 Distribution of Quadratic Forms 185......Page p185.djvu
3b.5 Three Fundamental Theorems of the Least Squares Theory 189......Page p189.djvu
3b.6 The p-Variate Normal Distribution 194......Page p194.djvu
3b.7 The Exponential Family of Distributions 195......Page p195.djvu
3c.2 Sampling Distributions 197......Page p197.djvu
3c.1 Definition 196......Page p196.djvu
3d.1 General Properties 201......Page p201.djvu
3d.2 Sampling Distributions 203......Page p203.djvu
Complements and Problems 209......Page p209.djvu
References 218......Page p218.djvu
4 The Theory of Least Squares and Analysis of Variance 220......Page p220.djvu
4a.1 Gauss-Markoff Setup (Y, X beta, sigma^2 I) 221......Page p221.djvu
4a.2 Normal Equations and Least Squares (l.s.) Estimators 222......Page p222.djvu
4a.3 g-Inverse and a Solution of the Normal Equation 224......Page p224.djvu
4a.4 Variances and Covariances of l.s. Estimators 226......Page p226.djvu
4a.5 Estimation of sigma^2 227......Page p227.djvu
4a.6 Other Approaches to the l.s. Theory (Geometric Solution) 228......Page p228.djvu
4a.7 Explicit Expressions for Correlated Observations 229......Page p229.djvu
4a.9 Least Squares Estimation with Restrictions on Parameters 231......Page p231.djvu
4a.10 Simultaneous Estimation of Parametric Functions 233......Page p233.djvu
4a.11 Least Squares Theory when the Parameters Are Random Variables 234......Page p234.djvu
4a.12 Choice of the Design Matrix 235......Page p235.djvu
4b Tests of Hypotheses and Interval Estimation 236......Page p236.djvu
4b.1 Single Parametric Function (Inference) 237......Page p237.djvu
4b.2 More than One Parametric Function (Inference) 238......Page p238.djvu
4b.3 Setup with Restrictions 242......Page p242.djvu
4c.2 Asymmetry of Right and Left Femora (Paired Comparison) 243......Page p243.djvu
4d.1 The Test Criterion 244......Page p244.djvu
4d.2 An Example 246......Page p246.djvu
4e.1 Single Observation in Each Cell 247......Page p247.djvu
4e.2 Multiple but Equal Numbers in Each Cell 252......Page p252.djvu
4e.3 Unequal Numbers in Cells 254......Page p254.djvu
4f.2 Variance Components Model 258......Page p258.djvu
4f.3 Treatment of the General Model 259......Page p259.djvu
4g.1 Concept of Regression (General Theory) 263......Page p263.djvu
4g.2 Measurement of Additional Association 268......Page p268.djvu
4g.3 Prediction of Cranial Capacity (a Practical Example) 270......Page p270.djvu
4g.4 Test for Equality of the Regression Equations 281......Page p281.djvu
4g.5 The Test for an Assigned Regression Function 284......Page p284.djvu
4g.6 Restricted Regression 287......Page p287.djvu
4h.1 Concomitant Variables 288......Page p288.djvu
4h.2 Analysis of Covariance 289......Page p289.djvu
4h.3 An Illustrative Example 291......Page p291.djvu
4i.1 A Basic Lemma on Generalized Inverse 294......Page p294.djvu
4i.2 The General Gauss-Markoff Model (GGM) 297......Page p297.djvu
4i.3 The Inverse Partitioned Matrix (IPM) Method 298......Page p298.djvu
4i.4 Unified Theory of Least Squares 300......Page p300.djvu
4j.1 Variance Components Model 302......Page p302.djvu
4j.2 MINQUE Theory 303......Page p303.djvu
4j.3 Computation under the Euclidian Norm 304......Page p304.djvu
4k.1 Best Linear Estimator (BLE) 305......Page p305.djvu
4k.2 Best Linear Minimum Bias Estimation (BLIMBE) 306......Page p306.djvu
Complements and Problems 308......Page p308.djvu
References 312......Page p312.djvu
5 Criteria and Methods of Estimation 314......Page p314.djvu
5a.1 Minimum Variance Criterion 315......Page p315.djvu
5a.2 Some Fundamental Results on Minimum Variance Estimation 317......Page p317.djvu
5a.3 The Case of Several Parameters 326......Page p326.djvu
5a.4 Fisher\'s Information Measure 329......Page p329.djvu
5a.5 An Improvement of Unbiased Estimators 332......Page p332.djvu
5b.2 Joint d.f of (theta, x) Completely Known 334......Page p334.djvu
5b.4 Empirical Bayes Estimation Procedures 336......Page p336.djvu
5b.5 Fiducial Probability 339......Page p339.djvu
5b.6 Minimax Principle 340......Page p340.djvu
5b.7 Principle of Invariance 343......Page p343.djvu
5c.1 Consistency 344......Page p344.djvu
5c.2 Efficiency 346......Page p346.djvu
5d.1 Method of Moments 351......Page p351.djvu
5d.2 Minimum Chi-Square and Associated Methods 352......Page p352.djvu
5d.3 Maximum Likelihood 353......Page p353.djvu
5e.1 Nonparametric Case 355......Page p355.djvu
5e.2 Parametric Case 359......Page p359.djvu
5f.1 Assumptions and Notations 363......Page p363.djvu
5f.2 Properties of m.l. Equation Estimators 364......Page p364.djvu
5g The Method of Scoring for the Estimation of Parameters 366......Page p366.djvu
Complements and Problems 374......Page p374.djvu
References 380......Page p380.djvu
6a.1 Asymptotic Distribution of Quadratic Functions of Frequencies 382......Page p382.djvu
6a.2 Some Convergence Theorems 385......Page p385.djvu
6b Chi-Square Tests for the Multinomial Distribution 390......Page p390.djvu
6b.2 Chi-Square Test for Goodness of Fit 391......Page p391.djvu
6b.3 Test for Deviation in a Single Cell 393......Page p393.djvu
6b.4 Test Whether the Parameters Lie in a Subset 395......Page p395.djvu
6b.5 Some Examples 396......Page p396.djvu
6b.6 Test for Deviations in a Number of Cells 397......Page p397.djvu
6c.1 General Results 398......Page p398.djvu
6c.2 Test of homogeneity of Parallel Samples 399......Page p399.djvu
6c.3 An Example 401......Page p401.djvu
6d.1 The Probability of an Observed Configuration and Tests in Large Samples 403......Page p403.djvu
6d.2 Tests of Independence in a Contingency Table 404......Page p404.djvu
6d.3 Tests of Independence in Small Samples 412......Page p412.djvu
6e.1 Notations and Basic Results 415......Page p415.djvu
6e.2 Test of a Simple Hypothesis 417......Page p417.djvu
6e.3 Test of a Composite Hypothesis 418......Page p418.djvu
6f.1 The Empirical Distribution Function 420......Page p420.djvu
6f.2 Asymptotic Distribution of Sample Fractiles 422......Page p422.djvu
6g.2 Square Root Transformation of the Poisson Variate 426......Page p426.djvu
6g.3 Sin^-1 Transformation of the Square Root of the Binomial Proportion 427......Page p427.djvu
6g.4 Tanh^-1 Transformation of the Correlation Coefficient 432......Page p432.djvu
6h.1 Variances and Covariances of Raw Moments 436......Page p436.djvu
6h.2 Asymptotic Variances and Covariances of Central Moments 437......Page p437.djvu
6h.3 Exact Expressions for Variances and Covariances of Central Moments 438......Page p438.djvu
Complements and Problems 439......Page p439.djvu
References 442......Page p442.djvu
7 Theory of Statistical Inference 444......Page p444.djvu
7a.1 Statement of the Problem 445......Page p445.djvu
7a.2 Neyman-Pearson Fundamental Lemma and Generalizations 446......Page p446.djvu
7a.3 Simple H_0 against Simple H 448......Page p448.djvu
7a.4 Locally Most Powerful Tests 453......Page p453.djvu
7a.5 Testing a Composite Hypothesis 456......Page p456.djvu
7a.6 Fisher-Behrens Problem 463......Page p463.djvu
7a.7 Asymptotic Efficiency of Tests 464......Page p464.djvu
7b.2 A General Method of Constructing a Confidence Set 470......Page p470.djvu
7b.3 Set Estimators for Functions of theta 473......Page p473.djvu
7c.1 Wald\'s Sequential Probability Ratio Test 474......Page p474.djvu
7c.2 Some Properties of the S.P.R.T. 475......Page p475.djvu
7c.3 Efficiency of the S.P.R.T. 478......Page p478.djvu
7c.4 An Example of Economy of Sequential Testing 480......Page p480.djvu
7c.5 The Fundamental Identity of Sequential Analysis 482......Page p482.djvu
7c.6 Sequential Estimation 485......Page p485.djvu
7c.7 Sequential Tests with Power One 488......Page p488.djvu
7d.2 Randomized and Nonrandomized Decision Rules 491......Page p491.djvu
7d.3 Bayes Solution 492......Page p492.djvu
7d.4 Complete Class of Decision Rules 495......Page p495.djvu
7d.5 Minimax Rule 496......Page p496.djvu
7e.1 Concept of Robustness 497......Page p497.djvu
7e.2 Distribution-Free Methods 499......Page p499.djvu
7e.3 Some Nonparametric Tests 500......Page p500.djvu
7e.4 Principle of Randomization 501......Page p501.djvu
7f Ancillary Information 505......Page p505.djvu
Complements and Problems 506......Page p506.djvu
References 512......Page p512.djvu
8 Multivariate Analysis 516......Page p516.djvu
8a.1 Definition 517......Page p517.djvu
8a.2 Properties of the Distribution 519......Page p519.djvu
8a.3 Some Characterizations of N_p 525......Page p525.djvu
8a.4 Density Function of the Multivariate Normal Distribution 527......Page p527.djvu
8a.5 Estimation of Parameters 528......Page p528.djvu
8a.6 N_p as a Distribution with Maximum Entropy 532......Page p532.djvu
8b.1 Definition and Notation 533......Page p533.djvu
8b.2 Some Results on Wishart Distribution 535......Page p535.djvu
8c.1 The Gauss-Markoff Setup for Multiple Measurements 543......Page p543.djvu
8c.2 Estimation of Parameters 544......Page p544.djvu
8c.3 Tests of Linear Hypotheses, Analysis of Dispersion (A.D.) 547......Page p547.djvu
8c.4 Test for Additional Information 551......Page p551.djvu
8c.5 The Distribution of Lambda 555......Page p555.djvu
8c.6 Test for Dimensionality (Structural Relationship) 556......Page p556.djvu
8c.7 Analysis of Dispersion with Structural Parameters (Growth Model) 561......Page p561.djvu
8d.1 Test for Assigned Mean Values 562......Page p562.djvu
8d.2 Test for a Given Structure of Mean Values 564......Page p564.djvu
8d.3 Test for Differences between Mean Values of Two Populations 565......Page p565.djvu
8d.4 Test for Differences in Mean Values between Several Populations 569......Page p569.djvu
8d.5 Barnard\'s Problem of Secular Variations in Skull Characters 572......Page p572.djvu
8e.1 Discriminant Scores for Decision 574......Page p574.djvu
8e.2 Discriminant Analysis in Research Work 577......Page p577.djvu
8e.3 Discrimination between Composite Hypotheses 580......Page p580.djvu
8f.1 Canonical Correlations 582......Page p582.djvu
8f.2 Properties of Canonical Variables 584......Page p584.djvu
8f.3 Effective Number of Common Factors 585......Page p585.djvu
8g.1 The Gram-Schmidt Basis 587......Page p587.djvu
8g.2 Principal Component Analysis 590......Page p590.djvu
Complements and Problems 593......Page p593.djvu
References 601......Page p601.djvu
Publications of the Author 605......Page p605.djvu
Author Index 615......Page p615.djvu
Subject Index 618......Page p618.djvu