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Lectures on Dependency

Part of the SpringerBriefs in Statistics Book Series

DOI: 10.1007/978-3-030-96932-5

ISBN: 978-3-030-96931-8

Publisher: Springer International Publishing - Springer Journals

Mar 12, 2022

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General Preliminaries

Dickhaus, Thorsten

Pages: 1 –8

[In this introductory chapter, some general notions and results are introduced and discussed. In particular, we discuss properties of covariances, correlations, copula functions, and normal distributions. Also some remarks regarding statistical tests are made. Finally, a conceptual overview of...
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Correlation Coefficients of Bivariate Normal Distributions

Dickhaus, Thorsten

Pages: 9 –18

[This lecture deals with the relationship between Pearson’s product-moment correlation coefficient and Spearman’s rank correlation coefficient under normal distributions (and more general elliptical distributions). This chapter also includes a section on Gaussian integrals of quadratic forms,...
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Empirical Likelihood Ratio Tests for the Population Correlation Coefficient

Dickhaus, Thorsten

Pages: 19 –23

[In this chapter, empirical likelihood ratio tests for Pearson’s product-moment correlation coefficient are discussed. As a side result, we also prove a nonparametric maximum likelihood estimator property of the empirical measure.]
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The Rearrangement Algorithm

Dickhaus, Thorsten

Pages: 25 –30

[In this lecture, we discuss certain optimization problems related to copula theory. A numerical solution to these problems is presented, namely, the so-called rearrangement algorithm.]
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On the Covariances of Order Statistics

Dickhaus, Thorsten

Pages: 31 –35

[This lecture deals with properties of order statistics of stochastically independent and identically distributed real-valued observables. In particular, it is shown that such order statistics are non-negatively correlated, and some corollaries are derived from this property. The chapter also...
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On Equi-Correlation Matrices

Dickhaus, Thorsten

Pages: 37 –41

[In this lecture, we analyze properties of equi-correlation matrices. In particular, we apply the Sherman-Morrison formula to the problem of inverting an equi-correlation matrix, and we provide a lower bound on the equi-correlation coefficient. Some more general remarks about matrix theory for...
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Skew-Normal Distributions

Dickhaus, Thorsten

Pages: 43 –48

[This lecture deals with skew-normal distributions on the real line. Utilizing results from stationary time series analysis, we derive a stochastic representation of a skew-normally distributed random variable. Furthermore, possible generalizations of skew-normal distributions are mentioned.]
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The Weighted Bootstrap

Dickhaus, Thorsten

Pages: 49 –53

[In this chapter, (weighted) bootstrap methods for testing statistical functionals are presented. The relationship to dependency is that Efron’s bootstrap involves multinominal weights which are correlated. Some modifications of these weights are proposed to address this issue.]

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