  We introduce and analyse a new nonparametric estimator of a multi-dimensional
density. Our smooth projection estimator (SPE) is defined by a least squares
projection of the sample onto an infinite dimensional mixture class via an
undersmoothed nonparametric pilot estimate, which acts as a structural filter
to regularise the solution. The undersmoothing is required to optimise the
convergence rate of the SPE, which is jointly determined by that of the pilot
estimator to the true density in squared $\mathbb{L}_{2}$ norm, and by that of
the pilot distribution function to the empirical distribution function in
uniform norm. Our procedure was conceived with a view to exploiting well known
results in convex analysis and their connection to mixture densities. In the
context of our work, this translates to the observation that the infinite
dimensional minimisation problem, implicit in the construction of the SPE,
possesses a solution of dimension at most $n+1$, where $n$ is the sample size.
The SPE thus enjoys practical advantages such as computational efficiency, ease
of storage and rapid evaluation at a new data point.

Methodology