Detection of Change-Points in Dependence Structures
Anne-Catherine Favre1,4,5, Jean-François Quessy2, Mériem Saïd1, Maryse Champagne3
1Département de Mathématiques et et de Statistique, Université Laval, Québec, QC, Canada; 2Département de Mathématiques et d'Informatique, Université du Québec à Trois-Rivières, Trois-Rivières, QC, Canada; 3Statistique Canada, Ottawa, ON, Canada; 4Ense3, Institut national polytechnique de Grenoble, Isère, France; 5Laboratoire d'étude des Transferts en Hydrologie et Environnement (LTHE), Grenoble, Isère, France

In this study, tests for the detection of change-points in the dependence structure are proposed. So far, the only available procedure for that problem is the test of Dias and Embrechts (2009). However, this method is parametric in the sense that the dependence structure is supposed to belong to a given parametric family and the margins are assumed to be known. These assumptions are often unrealistic in practice. The statistics proposed in this work assume nothing about the form of the underlying distributions. The starting point is the possibility to extract the dependence structure of a multivariate distribution, i.e. its copula. The procedures proposed in this work are based on Kendall's measure of association. This dependence index is attractive in our context since its population value depends only on the underlying copula of a random couple. Moreover, its sample version is easy to compute and a quick, valid re-sampling procedure is available for inference purposes. The test statistics that will be investigated in this work are L1-, L2-, and L-distances of an empirical process defined as differences of Kendall's measures of association. Their asymptotic distributions are derived and the validity of a re-sampling procedure, based on a non parametric boostrap, to compute p-values is established. The method is shown to be valid both asymptotically and for moderate sample sizes. Monte Carlo simulations show that the tests are powerful under many scenarios of change-point based on copulas. Also, an estimator of the time of change is proposed and its efficiency is investigated. We illustrate the proposed statistical tests with climatic data, more precisely with long-run simulations issued from the Canadian Regional Climate Model. We focus on precipitation and runoff for small watersheds located in the northern part of the province of Quebec (Canada).

Keywords: Change-point; Copula; Kendall's tau; Climate change

Biography: Anne-Catherine Favre obtained her Ph.D. in statistical hydrology at the Swiss Federal Institute of Technology in Lausanne in 2001. From 2002 to 2009, she was professor of statistics at INRS-ETE, Université du Québec. In September 2004, she was named to the NSERC Industrial Research Chair in Statistical Hydrology. She is now professor at the department of mathematics and statistic at Laval University. From January 2011, she will be professor at the engineering school on Energy, Water and Environment (ENSE3) of the Grenoble Institute of Technolog (Grenoble INP). She will conduct her research with the Laboratory of study of Transfers in Hydrology and Environment (LTHE). She is interested in statistics applied to hydrology and climatology, namely in calibration of meteorological and hydrological ensemble forecasts, in multivariate modeling with copulas, in the estimation of change-points in climatological simulations, in spatial modeling for extremes and in stochastic modeling of precipitation.