Dickhaus T. Lectures on Dependency: Selected Topics in Multivariate Statistics

Springer, 2022. — 68 p.This short book elaborates on selected aspects of stochastic-statistical dependencies in multivariate statistics. Each chapter provides a rigorous and self-contained treatment of one specific topic, poses a particular problem within its scope, and concludes by presenting its solution. The presented problems are not only relevant for research in mathematical statistics, but also entertaining, with elegant proofs and appealing solutions. The chapters cover correlation coefficients of bivariate normal distributions, empirical likelihood ratio tests for the population correlation, the rearrangement algorithm, covariances of order statistics, equi-correlation matrices, skew-normal distributions and the weighted bootstrap. This book is primarily intended for early-career researchers in mathematical statistics, but will also be interesting for lecturers in the field. Its goal is to rouse the reader’s interest, further their knowledge of the subject and provide them with some useful mathematical techniques.Preface
References
Acknowledgements
Acronyms
General Preliminaries
Covariances and Correlations
Copula Functions
Normal Distributions
Statistical Tests
Overview of the Remaining Chapters
References
Correlation Coefficients of Bivariate Normal Distributions
Introduction and Motivation
Gaussian Integrals of Quadratic Forms
Elementary Proof of Part (ii) of Theorem 2.2
More Elegant Proof of Part (ii) of Theorem 2.2
Extension to Kendall's tau
References
Empirical Likelihood Ratio Tests for the Population Correlation Coefficient
Introduction and Motivation
Empirical Likelihood Ratio Tests for Multivariate Means
Profile ELR Test for the population correlation coefficient
References
The Rearrangement Algorithm
Introduction
Numerical Solution of the Problem
Application to the Product of Marginally Uniform Random Variables
References
On the Covariances of Order Statistics
Introduction and Motivation
Verification of Non-negative Correlations Among the Order Statistics
References
On Equi-Correlation Matrices
Introduction and Motivation
Computing the Inverse of an Equi-Correlation Matrix
Lower Bound on the equi-correlation parameter via the Determinant of the equi-correlation matrix
Matrix Theory for Statisticians
References
Skew-Normal Distributions
Introduction and Motivation
Stochastic Representation of the Skew-Normal Distribution
Generalizations
References
The Weighted Bootstrap
Introduction: Testing Statistical Functionals
Efron's Bootstrap
Weighted Bootstrap Procedures
References
Appendix Index
Index