The present article is devoted to the semi-parametric estimation of multivariate expectiles for extreme levels. The considered multivariate risk measures also include the possible conditioning with respect to a functional covariate, belonging to an infinite-dimensional space. By using the first order optimality condition, we interpret these expectiles as solutions of a multidimensional nonlinear optimum problem. Then the inference is based on a minimization algorithm of gradient descent type, coupled with consistent kernel estimations of our key statistical quantities such as conditional quantiles, conditional tail index and conditional tail dependence functions. The method is valid for equivalently heavy-tailed marginals and under a multivariate regular variation condition on the underlying unknown random vector with arbitrary dependence structure. Our main result establishes the consistency in probability of the optimum approximated solution vectors with a speed rate. This allows us to estimate the global computational cost of the whole procedure according to the data sample size.

翻译：本文致力于极端水平下半参数多元期望值的估计。所考虑的多元风险度量还包括可能存在的、关于属于无限维空间的函数协变量的条件作用。通过利用一阶最优性条件，我们将这些期望值解释为多维非线性最优问题的解。随后，基于梯度下降类型的极小化算法进行推断，并结合对我们的关键统计量（如条件分位数、条件尾指数和条件尾相依函数）的一致核估计。该方法适用于等重尾边际分布，并且基于具有任意相依结构的潜在未知随机向量满足多元正则变化条件。我们的主要结果建立了具有速度率的最优近似解向量的概率相合性。这使我们能够根据数据样本量估计整个过程的全局计算成本。