Extremes with random covariates

Classical extreme value statistics assumes that observations are independent and identically distributed. In this case, the tail distribution of the focal variable is approximated by a Generalized Pareto Distribution (GPD). In applications where the tail distributions may vary according to covariates, the parameters of the GPD are often modeled by parametric functions of the covariates. In this study, we introduce the \textit{proportional tail model}, which extends the heteroskedastic extremes model in Einmahl et al. (2016) by incorporating multiple random covariates into the tail distribution based on generic parametric models. We provide theoretical justifications for a likelihood based estimation of such parametric models. Einmahl et al. (2016) introduces the scedasis function to model the scale variation in heavy-tailed distributions with respect to a single deterministic covariate, time. Mefleh et al (2020) models the scedasis as a parametric function of time. By contrast, the \textit{proportional tail model} allows for negative extreme value index with endpoints depending on multiple random covariates. We show