  We consider Bayesian nonparametric density estimation using a Pitman-Yor or a
normalized inverse-Gaussian process kernel mixture as the prior distribution
for a density. The procedure is studied from a frequentist perspective. Using
the stick-breaking representation of the Pitman-Yor process or the expression
of the finite-dimensional distributions for the normalized-inverse Gaussian
process, we prove that, when the data are replicates from an infinitely smooth
density, the posterior distribution concentrates on any shrinking $L^p$-norm
ball, $1\leq p\leq\infty$, around the sampling density at a \emph{nearly
parametric} rate, up to a logarithmic factor. The resulting hierarchical
Bayesian procedure, with a fixed prior, is thus shown to be adaptive to the
infinite degree of smoothness of the sampling density.
