Dimension reduction for the estimation of the conditional tail index

We are interested in the relationship between the large values of a real random variable and its associated multidimensional covariate, in the context where the conditional distribution is heavy‐tailed. Estimating the positive conditional tail index of a heavy‐tailed conditional distribution is a crucial step for statistical inference, but the task becomes increasingly challenging as the covariate dimension increases. In this work, we assume the existence of a lower‐dimensional linear subspace such that the conditional tail index depends on the covariate only through its projection onto this subspace. We propose a method to estimate this dimension reduction subspace and establish its consistency. Additionally, we introduce an estimator of the conditional tail index that leverages this dimension reduction and prove its consistency. We illustrate the benefits of this dimension reduction approach for estimating the conditional tail index through simulations and an application to real‐world data.