Multivariate Analysis—III

List of ContributorsPrefaceAcknowledgmentsPart I/Time Series and Stochastic Processes Two-Dimensional Random Fields Preliminaries Details of the Proof of Theorem 1 References Concepts of Consistency in Spectral Estimation for Multivariate Time Series 1. Introductory Remarks 2. Should One Estimate the Spectral Distribution Function or Its Derivative? 3. Estimation of the Spectral Distribution Function F for a Multivariate Stationary Random Sequence Appendix References Non-Anticipative Canonical Representations of Equivalent Gaussian Processes 1. Introduction 2. The General Form of the Non-Anticipative Representation 3. A Derivation Using Martingale Theory 4. Concluding Remarks References Abstract Martingales and Ergodic Theory Introduction 1. The Problem 2. Martingale Formulation 3. An Operator Theoretic Approach 4. A Maximal Inequality 5. Final Remarks References On the Modelling and Estimation of Communication Channels 1. Introduction and Preliminary Discussion 2. Classes of Channels and Representations 3. Estimation of Parameters in a Linear Model 4. Channel Identification 5. Remarks References Innovation and Nonanticipative Processes 1. Innovation Processes and Regularity 2. Canonical Representations and Fully Submitted Processes ReferencesPart II/Distribution Theory and Inference Methods for Assessing Multivariate Normality 1. Introduction 2. Univariate Techniques for Evaluating Marginal Normality 3. Multivariate Techniques for Evaluating Joint Normality 4. Tests Based on Unidimensional Views of Multivariate Data 5. Examples 6. Concluding Remarks References Asymptotic Expansions for the Distributions of Characteristic Roots When the Parameter Matrix Has Several Multiple Roots 1. Introduction 2. The Maximization Procedures 3. Asymptotic Expansion for the Distribution of the Latent Roots of the Estimated Covariance Matrix—Several Multiple Population Roots 4. Asymptotic Expansion for the Distribution of the Latent Roots of S1 S21—Several Multiple Population Roots 5. Asymptotic Expansion for Manova—Several Multiple Population Roots 6. Asymptotic Expansion for Canonical Correlation—Several Multiple Population Roots 7. Complex Analogues of Previous Results 8. Remarks References Aspects of the Multinomial Logit Model 1. The General Logit Model 2. Properties of the Likelihood 3. Comment Appendix: The Beaton Sweep References Inference and Redundant Parameters 1. Introduction 2. The Probability Space Model 3. Measure Factorizations 4. If the Inner Parameter Become Known 5. A Redundant Parameter 6. The Multivariate Model 7. The Bayesian Right Invariant References The Variance Information Manifold and the Functions on It 1. The Variance Information Manifold and Its Boundary 2. The Bivariate Case 3. The Multinormal Distribution with Singular Information Matrix 4. Derivation via the Distribution of Linear Functions 5. Application to the Analysis of Experimental Designs 6. Representations as the Marginal Distribution of a Nonsingular Distribution 7. Decomposition of a Multinormal Distribution 8. Invariant Metric 9. Geodesic Distance between Two Matrices 10. Zonal Polynomials References Stopping Time in Sequential Samples from Multivariate Exponential Families 1. Introduction 2. The Main Theorem 3. Application to Examples 1.1 and 1.2 ReferencesPart III/Characteristic Functions and Characterizations An Isomorphism Method for the Study of I0n 1. Introduction 2. Definitions and Notations 3. Isomorphism Method 4. Applications in the General Case 5. Applications to a Finite Independent Set 6. Applications to An Enumerable Independent Set 7. Finite Products of Poisson Laws 8. α-Decompositions References A Characterization of the Multivariate Geometric Distribution 1. Introduction 2. A Regression Property 3. The Characterization Theorem 4. Derivation of the Differential Equations 5. Completion of the Proof of Sufficiency 6. Proof of the Necessity References On Infinitely Decomposable Probability Distributions, and Helical Varieties in Hilbert Space 1. Introduction 2. The Canonical Association of Helical Varieties with Infinitely Decomposable Distributions 3. A Hilbert Space Proof of the Lévy-Khinchine Theorem for R0 4. Operator-Measure Theoretic Treatment 5. On Probability and Hilbert Spaces 6. Bibliographical and Concluding Remarks References Limit Laws for Sequences of Normed Sums Satisfying Some Stability Conditions ReferencesPart IV/Design and Analysis of Experiments The Analysis of Time Series Collected in an Experimental Design 1. Introduction 2. The Finite Fourier Transform 3. The Fixed Effects Model 4. The Random Effects Model 5. The Point Process Case Appendix References Max-Min Designs in the Analysis of Variance 1. Introduction and Max-Min Designs 2. A Matrix Inequality and Regular Designs References Analysis of Covariance Structures 1. Introduction 2. General Results 3. Applications References Optimum Designs for Fitting Biased Multiresponse Surfaces 1. Introduction 2. Formulation of Box and Draper 3. Minimizing V + B 4. An Illustrative Example 5. Other Comments References Asymptotic Properties of Some Sequential Nonparametric Estimators in Some Multivariate Linear Models 1. Introduction 2. The Problems 3. Preliminary Notions and Basic Assumptions 4. Asymptotic Properties of Robust Sequential Point Estimators of (α,β) 5. Bounded Length (Sequential) Confidence Bands for θ ReferencesPart V/Classification, Modelling, and Reliability Availability Theory for Multicomponent Systems Introduction and Summary 1. Preliminaries 2. Average System Up Time : Almost Sure Results 3. Asymptotic Distributions 4. Cost of Repair References Some Measures for Discriminating Between Normal Multivariate Distributions with Unequal Covariance Matrices 1. Summary and Introduction 2. The Measure S 3. The Measure T 4. The Kullback-Leibler Information Numbers References Correlation and Affinity in Gaussian Cases 1. Introduction 2. Correlation Coefficient and ρI ρII 3. Canonical Correlation and ρI or ρII References Identification of the Structure of Multivariable Stochastic Systems 1. Introduction 2. Two Results in Principal Components Analysis 3. Multivariable Linear Systems 4. Identification of the System Structure 5. Reduction of the Dimension of the Output Vector 6. Reduction of the Dimension of the Input Vector 7. Practical Problems 8. Tests of Significance of Eigenvalues 9. Testing the Equality of λk+1 (ω) = λk+2 (w) = ...λρ(ω) = λ(ω) 10. Asymptotic Theory for the Distribution of Eigenvalues References An Information Function Approach to Dimensionality Analysis and Curved Manifold Clustering 1. Introduction 2. Entropy in the Discrete and Continuous Cases 3. Uncertainty and Dimesionality 4. Some Auxiliary Techniques 5. Monte Carlo Studies References Nonlinear Iterative Partial Least Squares (NIPALS) Modelling: Some Current Developments 1. Introduction and Summary 2. What is NIPALS Modelling? 3. Low Information versus High Information Modelling 4. Causal Flow Models in Econometrics and the Behavioural Sciences 5. Case Studies in NIPALS Modelling ReferencesTitles of Contributed Papers