We study minimax risk bounds for estimators of the spectral measure in multivariate linear factor models, where observations are linear combinations of regularly varying latent factors. Non-asymptotic convergence rates are derived for the multivariate Peak-over-Threshold estimator in terms of the $p$-th order Wasserstein distance, and information-theoretic lower bounds for the minimax risks are established. The convergence rate of the estimator is shown to be minimax optimal under a class of Pareto-type models analogous to the standard class used in the setting of one-dimensional observations known as the Hall-Welsh class. When the estimator is minimax inefficient, a novel two-step estimator is introduced and demonstrated to attain the minimax lower bound. Our analysis bridges the gaps in understanding trade-offs between estimation bias and variance in multivariate extreme value theory.

翻译：我们研究了多变量线性因子模型中谱测度估计量的极小极大风险界，其中观测值由规则变化的潜在因子线性组合而成。针对多变量峰值超过阈值估计量，建立了基于p阶Wasserstein距离的非渐近收敛速率，并给出了风险下界的信息论结果。在类似标准帕累托模型类（即一维观测中常用的Hall-Welsh类）的框架下，该估计量的收敛速率被证明具有极小极大最优性。当估计量非极小极大有效时，我们引入了一种新颖的两步估计方法，并证明其能达到极小极大下界。本分析弥补了多变量极值理论中估计偏差与方差权衡理解上的空白。