This book is a collaborative effort from three workshops held over the last three years, all involving principal contributors to the vine-copula methodology. Research and applications in vines have ...been growing rapidly and there is now a growing need to collate basic results, and standardize terminology and methods. Specifically, this handbook will (1) trace historical developments, standardizing notation and terminology, (2) summarize results on bivariate copulae, (3) summarize results for regular vines, and (4) give an overview of its applications. In addition, many of these results are new and not readily available in any existing journals. New research directions are also discussed.
The influence of information about the dependence of random variables on the approximation properties of a nonparametric estimate of the probability density of the Rosenblatt–Parzen type is ...determined. The ratio of the asymptotic expressions of the mean square deviations of independent and dependent random variables is obtained. This ratio for two-dimensional random variables is considered as a quantitative estimate of the influence of information about their dependence on the approximation properties of the kernel estimate of the probability density. The established ratio is determined by the type of the probability density and the volumes of the initial statistical data, which are used to estimate the probability densities of dependent and independent random variables. The general results obtained are considered in detail in relation to two-dimensional linearly dependent random variables with normal distribution laws. The functional dependence of the ratio of standard deviations of independent and dependent two-dimensional random variables on the correlation coefficient is determined and the dependence of this ratio on the volume of statistical data is analyzed. A method for estimating the functional of the second derivatives of two-dimensional random variables with normal distribution laws has been developed. The results obtained are the basis for modifying "fast" procedures for optimizing kernel estimates of probability densities in conditions of large samples.
A simple measure of similarity for the construction of the market graph is proposed. The measure is based on the probability of the coincidence of the signs of the stock returns. This measure is ...robust, has a simple interpretation, is easy to calculate and can be used as measure of similarity between any number of random variables. For the case of pairwise similarity the connection of this measure with the sign correlation of Fechner is noted. The properties of the proposed measure of pairwise similarity in comparison with the classic Pearson correlation are studied. The simple measure of pairwise similarity is applied (in parallel with the classic correlation) for the study of Russian and Swedish market graphs. The new measure of similarity for more than two random variables is introduced and applied to the additional deeper analysis of Russian and Swedish markets. Some interesting phenomena for the cliques and independent sets of the obtained market graphs are observed.
The concept of dependence permeates the Earth and its inhabitants in a most profound manner. Examples of interdependent meteorological phenomena in nature and interdependence in the medical, social, ...and political aspects of our existence, not to mention the economic structures, are too numerous to be cited individually. Moreover, the dependence is obviously not deterministic but of a stochastic nature. However, it seems that none of the departments of statistics, engineering, economics and mathematics in the academic institutions throughout the world offer courses dealing with dependence concepts and measures.This book can thus be viewed as an attempt to remedy the situation, and it has been written for a graduate course or a seminar on correlation and dependence concepts and measures. A modest background in mathematical statistics and probability and integral calculus is required. The book is not a full-scale expedition up another statistical Alp. Rather, it is a tour over a somewhat neglected but important terrain. The chapter on correlation is written for a layman.
