  Current tools for multivariate density estimation struggle when the density
is concentrated near a nonlinear subspace or manifold. Most approaches require
choice of a kernel, with the multivariate Gaussian by far the most commonly
used. Although heavy-tailed and skewed extensions have been proposed, such
kernels cannot capture curvature in the support of the data. This leads to poor
performance unless the sample size is very large relative to the dimension of
the data. This article proposes a novel generalization of the Gaussian
distribution, which includes an additional curvature parameter. We refer to the
proposed class as Fisher-Gaussian (FG) kernels, since they arise by sampling
from a von Mises-Fisher density on the sphere and adding Gaussian noise. The FG
density has an analytic form, and is amenable to straightforward implementation
within Bayesian mixture models using Markov chain Monte Carlo. We provide
theory on large support, and illustrate gains relative to competitors in
simulated and real data applications.
