We study the problem of estimating a nonparametric probability density under
a large family of losses called Besov IPMs, which include, for example,
$\mathcal{L}^p$ distances, total variation distance, and generalizations of
both Wasserstein and Kolmogorov-Smirnov distances. For a wide variety of
settings, we provide both lower and upper bounds, identifying precisely how the
choice of loss function and assumptions on the data interact to determine the
minimax optimal convergence rate. We also show that linear distribution
estimates, such as the empirical distribution or kernel density estimator,
often fail to converge at the optimal rate. Our bounds generalize, unify, or
improve several recent and classical results. Moreover, IPMs can be used to
formalize a statistical model of generative adversarial networks (GANs). Thus,
we show how our results imply bounds on the statistical error of a GAN,
showing, for example, that GANs can strictly outperform the best linear
estimator.