Dimension reduction for the estimation of the conditional tail index
针对高维协变量下重尾条件分布的尾部指数估计难题，假设存在低维线性子空间使尾部指数仅依赖投影，提出降维子空间估计方法并证明一致性，通过模拟和实际数据验证效果。
ABSTRACT 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.