A Nonparametric Dependence Measure for Random Variables Based on the One-to-One Correspondence between Copulas and Markov Operators
Wolfgang Trutschnig
Research Unit for Intelligent Data Analysis and Graphical Models, European Centre for Soft Computing, Mieres, Spain

Using the one-to-one correspondence between copulas and Markov operators as pointed out in [2], [3] and expressing the Markov operators in terms of regular conditional distributions (Markov kernels) allows to define a new metric D on the space of all copulas. This metric is not only a metrization of strong operator convergence of the corresponding Markov operators (see [2], [3]) but it also induces a natural dependence measure with various good properties by simply defining the dependence of a copula A as the D-distance between A and the product copula. In particular it can be shown that the class of copulas that have maximum D-distance to the product copula is exactly the class of all deterministic copulas (i.e. copulas that are induced by a Lebesgue-measure preserving transformation on [0,1]). Expressed in terms of the dependence measure this means that exactly deterministic copulas are assigned maximum dependence. As a consequence, the product copula can not be approximated arbitrarily well by deterministic copulas with respect to the metric D - this is quite the contrary when considering Schweizer and Wolff's sigma (see [1]) and the uniform convergence, in which case for instance the family of all shuffles of the minimum copula are dense in the space of all copulas (see [4]).

The main properties of metric D and the induced dependence measure will be presented and it will be shown that, although the metric D is quite strong, there exist various families of 'simple' copulas that can be used to approximate an arbritrary one.

References:

[1] B. Schweizer, E.F. Sklar: On nonparametric measures of dependence for random variables, Annals of Statistics, Volume 9, No. 4, 879-885 (1981)

[2] X. Li, P. Mikusinski, M.D. Taylor: Strong approximation of copulas, Journal of Mathematical Analysis and Applications, 255, 608-623 (1998)

[3] E.T. Olsen, W.F. Darsow, B. Nguyen: Copulas and Markov operators, Distributions with Fixed Marginals and Related Topics, IMS Lecture Notes, Monograph Series Vol. 28, 1996

[4] F. Durante, P. Sarkoci, C. Sempi: Shuffles of copulas, Journal of Mathematical Analysis and Applications, 352, 914-921 (2009)

Keywords: copula; double stochastic measure; dependence; Markov operator

Biography: Wolfgang Trutschnig received both his M.Sc. and Ph.D. in Mathematics (with emphasis in Statistics) from the Vienna University of Technology. During his Ph.D. studies he worked at the Statistics Department of the National Bank of Austria. In 2006 he was awarded the prize for dissertations in mathematical statistics from the Austrian Statistical Society. After two post-doc years at the Institute for Statistics and Probability Theory at the Vienna University of Technology he joined the European Centre for Soft Computing (ECSC) as post-doc researcher. Since January 2011 he is Associate researcher in the research unit for Intelligent Data Analysis and Graphical models at the ECSC.