Robust Density Estimation under Besov IPM Losses
This provides robust density estimation methods for statistical applications with contaminated data, but it is incremental as it extends existing results to a broader class of losses.
The paper tackles the problem of nonparametric density estimation in the Huber contamination model, where some data are outliers, and shows that a re-scaled thresholding wavelet series estimator achieves minimax optimal convergence rates under a family of Besov IPM losses, including L^p and Wasserstein distances.
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.