Interpretation of the measure of dependence for multivariate stable random variables using factor model
Vadim Shergin · Eastern-European Journal of Enterprise Technologies · 2015
The problem of interpretation the measures of dependence of multivariate stable random variables in the space of these variables was considered. It is noted that both the examined distribution laws and the existing measures of dependence are parameterized using characteristic functions, i.e. in the frequency domain, not in the space of random variables. This is what causes the relevance of the studied problem.The paper presents a model of symmetric mixing of hidden factors that allows to consider the dependence between random variables within the factor model. Under this approach, a set of identically distributed observed random variables is seen as the result of a linear transformation of the same number of independent random variables (hidden factors). It is shown that for the multivariate normal distribution, the correlation coefficient can be interpreted as a measure of the differences between the linear transformation of factors and identical transformation.The analysis of parameterization forms of multivariate stable distribution laws was performed. The subclass of such distributions, for which it is possible to represent the observed random variables by a linear combination of the independent variables was singled out. It is important to note that this subclass is parameterized unambiguously, in contrast to the general case of multivariate stable laws.It is shown that within the selected subclass of multivariate stable distributions, the measure of dependence has the same meaning as the correlation coefficient for the multivariate normal law.