# Multivariate Analysis—III

### Proceedings of the Third International Symposium on Multivariate Analysis Held at Wright State University, Dayton, Ohio, June 19-24, 1972

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Multivariate Analysis — III contains the proceedings of the Third International Symposium on Multivariate Analysis held at Wright State University in Dayton, Ohio, on June 19-24, 1972. The papers explore the theory and applications of multivariate analysis and cover areas such as time series and stochastic processes; distribution theory and inference; characteristic functions and characterizations; and design and analysis of experiments. Classification, modeling, and reliability are also discussed. Comprised of 27 chapters, this volume begins with an introduction to two-dimensional random fields, giving results for a class of Gaussian processes with a multidimensional time parameter. The next chapter deals with concepts of consistency in spectral estimation for multivariate time series and considers the alternative of estimating the spectral distribution function or the spectral density function. Abstract martingales and ergodic theory are also examined, along with methods for assessing multivariate normality; inference and redundant parameters; characterization of the multivariate geometric distribution; and max-min designs in the analysis of variance. This monograph will be useful to statisticians and probabilists, as well as to scientists in other disciplines who are broadly interested in multivariate analysis.

## Table of Contents

List of Contributors

Preface

Acknowledgments

Part 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

References

Part 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

References

Part 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

References

Part 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

References

Part 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

References

Titles of Contributed Papers

## Product details

- No. of pages: 428
- Language: English
- Copyright: © Academic Press 1973
- Published: January 1, 1973
- Imprint: Academic Press
- eBook ISBN: 9781483265131