  A theorem of McCann shows that for any two absolutely continuous probability
measures on R^d there exists a monotone transformation sending one probability
measure to the other. A consequence of this theorem, relevant to statistics, is
that density estimation can be recast in terms of transformations. In
particular, one can fix any absolutely continuous probability measure, call it
P, and then reparameterize the whole class of absolutely continuous probability
measures as monotone transformations from P. In this paper we utilize this
reparameterization of densities, as monotone transformations from some P, to
construct semiparametric and nonparametric density estimates. We focus our
attention on classes of transformations, developed in the image processing and
computational anatomy literature, which are smooth, invertible and which have
attractive computational properties. The techniques developed for this class of
transformations allow us to show that a penalized maximum likelihood estimate
(PMLE) of a smooth transformation from P exists and has a finite dimensional
characterization, similar to those results found in the spline literature.
These results are derived utilizing an Euler-Lagrange characterization of the
PMLE which also establishes a surprising connection to a generalization of
Stein's lemma for characterizing the normal distribution.
