Lectures on Dependency: Selected Topics in Multivariate Statistics (SpringerBriefs in Statistics)

✍ Scribed by Thorsten Dickhaus

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 64
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Preface
References
Acknowledgements
Contents
Acronyms
1 General Preliminaries
1.1 Covariances and Correlations
1.2 Copula Functions
1.3 Normal Distributions
1.4 Statistical Tests
1.5 Overview of the Remaining Chapters
References
2 Correlation Coefficients of Bivariate Normal Distributions
2.1 Introduction and Motivation
2.2 Gaussian Integrals of Quadratic Forms
2.3 Elementary Proof of Part (ii) of Theorem 2.2
2.4 More Elegant Proof of Part (ii) of Theorem 2.2
2.5 Extension to Kendall's tau
References
3 Empirical Likelihood Ratio Tests for the Population Correlation Coefficient
3.1 Introduction and Motivation
3.2 Empirical Likelihood Ratio Tests for Multivariate Means
3.3 Profile ELR Test for the population correlation coefficient
References
4 The Rearrangement Algorithm
4.1 Introduction
4.2 Numerical Solution of the Problem
4.3 Application to the Product of Marginally Uniform Random Variables
References
5 On the Covariances of Order Statistics
5.1 Introduction and Motivation
5.2 Verification of Non-negative Correlations Among the Order Statistics
References
6 On Equi-Correlation Matrices
6.1 Introduction and Motivation
6.2 Computing the Inverse of an Equi-Correlation Matrix
6.3 Lower Bound on the equi-correlation parameter via the Determinant of the equi-correlation matrix
6.4 Matrix Theory for Statisticians
References
7 Skew-Normal Distributions
7.1 Introduction and Motivation
7.2 Stochastic Representation of the Skew-Normal Distribution
7.3 Generalizations
References
8 The Weighted Bootstrap
8.1 Introduction: Testing Statistical Functionals
8.2 Efron's Bootstrap
8.3 Weighted Bootstrap Procedures
References
Appendix Index
Index