Joint PhD with CNRS: Modeling of multivariate data with complex dependence structures using copulas

Copula modeling has been applied in a wide variety of areas such as quantitative risk management, econometric modeling, environmental modeling, to name a few. Copulas arise as the functions which “couple together” the marginal cumulative distribution functions (c.d.f.s) of a random vector to form its joint c.d.f. From the point of view of the estimation of a multivariate distribution from data (a challenge that arises in all of the aforementioned application areas), this offers a great deal of flexibility as it allows practitioners to model the marginal c.d.f.s (that is, the univariate behaviours of the multivariate random phenomena of interest) separately from the dependence structure of the random phenomena (represented by the copula).

Within this setting, we have identified the following interesting research directions:

Sub-project 1 title: Spatial statistics and big data. Description: Copulas can lead to a natural extension of classical spatial statistics by replacing the underlying multivariate normal distribution by a copula. Recently, some flexible copula models have been proposed in the literature but the exact inference for these models is however not feasible in very high dimensions, for example, when modeling satellite image data or climate model output data. We plan to investigate fast approximate inference methods to tackle such big data sets with spatial dependence.

Sub-project 2 title: Model selection and diagnostics. Description: Model selection is crucial in statistics and has recently received the attention of specialists for copula-based models. Yet, several theoretical and computational aspects remain to be addressed such as taking into account the underlying uncertainty when estimating model scores to rank the candidate models. To do so, we plan to rely on resampling methods. The asymptotic validation of such computations arises as one of the main theoretical challenges of this research direction.

Sub-project 3 title: Measures of extremal dependence and model diagnostics. Description: Dependence measures and similarity measures can provide a useful summary of the dependence structure of high-dimensional data. These measures can be used as diagnostic tools to help find an appropriate copula model and can be used to develop goodness-of- t tests for multivariate models. We plan to develop measures of extremal dependence that can be used to assess the strength of dependence in the tails of a multivariate distribution (that is, when several variables take very large values). Possible applications include goodness-of- t tests for spatial extremes data such as monthly maximum wind speeds or extreme rainfall events.