Multivariate Return Periods and Hazard Measures: The Copula Approach
Gianfausto Salvadori1, Fabrizio Durante2
1Department of Mathematics, University of Salento, Lecce, Italy; 2School of Economics and Management, Free University of Bozen-Bolzano, Bolzano, Italy

The return period of a prescribed event is widely used in hydrology (as well as in geophysics) as a tool for design purposes, and as a means for risk assessment and rational decision making.

Given a prescribed return period, the engineering practice requires the identification of a critical design event. As a difference with the univariate case, in a multivariate setting an infinite number of events may have the same return period, a puzzling situation. This work is of methodological nature, and tries to make the point clear.

First, we prove that, in a multivariate context, the return period depends upon the Copula of the variables considered, and therefore its correct calculation could be carried out via the Kendall's measure.

Secondly, we show that the identification of critical design events can be carried out via a Maximum Likelihood approach, by calculating the most likely events over suitable critical layers of the probability density at play.

Finally, the implications of the results presented for risk assessment and rational decision making are illustrated and discussed.

Keywords: Multivariate return period; Multivariate design event; Copula modeling; Risk assessment

Biography: Gianfausto SALVADORI is an applied mathematician, presently researcher in Probability and Statistics at the University of Salento (Italy). He has been involved in environmental research since 1989, and has been concerned with several European projects regarding Chernobyl radioactive pollution and stochastic modeling via Universal Multifractals. Since 2001 he started working on Copulas, a mathematical tool for modeling multivariate dependent random variables. He also works with a team of hydrological engineers at the Polytechnic of Milan (Italy), and as a part of this team he has been involved in several national and European projects on extreme rainfall, floods, and sea storms. He recently co-published a book on Extremes and Copulas by Springer's (2007).