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

Wasserstein-based Minimax Estimation of Dependence in Multivariate Regularly Varying Extremes