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Statistics and applied probability provide the mathematical foundations for understanding complex dependence structures and stochastic dynamics. These tools are essential whenever uncertainty cannot be described adequately through marginal distributions alone, or when the evolution of a system over time depends on its current state.
My research in this area focuses on multivariate dependence modelling, in particular through hierarchical copula constructions, as well as on the statistical estimation of such models using computationally tractable methods such as composite likelihood. A complementary line of work concerns stochastic processes with state-dependent dynamics, where questions of recurrence, stability, and long-term behaviour play a central role.
The underlying objective is to develop probabilistic and statistical models that remain both mathematically rigorous and sufficiently flexible to represent complex real-world systems. These questions arise naturally in risk modelling, finance, insurance, reliability, and many other fields where dependence and dynamic uncertainty are key components of the analysis.
Chaoubi, I., Cossette, H., Marceau, E. and Robert, C. (2021). “Hierarchical copulas with Archimedean blocks and asymmetric between-block pairs.”
Computational Statistics & Data Analysis, 154, 107071.
Abstract
A new class of hierarchical copulas is introduced based on joint survival functions of multivariate exponential mixture distributions. The key element of this construction is the mixing random vector defined by convolutions associated to a Lévy subordinator, and leading to hierarchical copulas with Archimedean within-block copulas and asymmetric between-block pair-copulas. For the specific case of two-level trees, dependence properties of pairs are investigated, and a full estimation procedure is proposed for the tree structure and parameters of the hierarchical copulas. The efficiency of the procedure is illustrated through three simulation examples and a study with two real datasets.
Cossette, H., Gadoury, S.-P., Marceau, E. and Robert, C. (2019). “Composite likelihood estimation method for hierarchical Archimedean copulas defined with multivariate compound distributions.”
Journal of Multivariate Analysis, 172, 59–83.
Abstract
We consider the family of hierarchical Archimedean copulas obtained from multivariate exponential mixture distribution through compounding, as introduced by Cossette et al. (2017). We investigate ways of determining the structure of these copulas and estimating their parameters. An agglomerative clustering technique based on the matrix of Spearman’s rhos, combined with a bootstrap procedure, is used to identify the tree structure. Parameters are estimated through a top-down composite likelihood. The validity of the approach is illustrated through two simulation studies in which the procedure is explained step by step. The composite likelihood method is also compared to the full likelihood method in a simple case where the latter is computable.
Robert, C. (2007). “Stochastic stability of some state-dependent growth-collapse processes.”
Advances in Applied Probability, 39(1), 189–220.
DOI : 10.1239/aap/1175266475
Abstract
In this paper we consider a discrete-time process which grows according to a random walk with nonnegative increments between crash times at which it collapses to 0. We assume that the probability of crashing depends on the level of the process. We study the stochastic stability of this growth-collapse process. Special emphasis is given to the case in which the probability of crashing tends to 0 as the level of the process increases. In particular, we show that the process may exhibit long-range dependence and that the crash sizes may have a power law distribution.
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