  We study minimax convergence rates of nonparametric density estimation in the
Huber contamination model, in which a proportion of the data comes from an
unknown outlier distribution. We provide the first results for this problem
under a large family of losses, called Besov integral probability metrics
(IPMs), that includes $\mathcal{L}^p$, Wasserstein, Kolmogorov-Smirnov, and
other common distances between probability distributions. Specifically, under a
range of smoothness assumptions on the population and outlier distributions, we
show that a re-scaled thresholding wavelet series estimator achieves minimax
optimal convergence rates under a wide variety of losses. Finally, based on
connections that have recently been shown between nonparametric density
estimation under IPM losses and generative adversarial networks (GANs), we show
that certain GAN architectures also achieve these minimax rates.
