  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.
