Germain Van Bever: Interactions in statistics

In statistics, understanding interactions between variables, or equivalently the geometry of the underlying distribution, is essential to develop ad hoc methodologies. This is particularly true in the context of nonparametric statistics, where no particular shape (such as, for example, normality) is assumed for the distributions. In those settings, it is therefore necessary to study the complete geometry of the data rather than summarising it to a few parameters.

For one-dimensional variables, quantiles successfully capture the relevant information contained in the data, so that methods based on those will be “robust” to the misspecification of the model. These quantities do not have equivalent in the multivariate setting, precisely because they fail to study the interactions between variables.

Statistical depth functions study the global behaviour of the data by assessing the centrality of each observations within the whole dataset. Depth methods have become more and more popular over the last decades and illustrate the growing importance geometrical methods are taking in mathematical statistics. This talks aims at showcasing several new depth-based notions in various statistical problems.