  While the problem of estimating a probability density function (pdf) from its
observations is classical, the estimation under additional shape constraints is
both important and challenging. We introduce an efficient, geometric approach
for estimating pdfs given the number of its modes. This approach explores the
space of constrained pdf's using an action of the diffeomorphism group that
preserves their shapes. It starts with an initial template, with the desired
number of modes and arbitrarily chosen heights at the critical points, and
transforms it via: (1) composition by diffeomorphisms and (2) normalization to
obtain the final density estimate. The search for optimal diffeomorphism is
performed under the maximum-likelihood criterion and is accomplished by mapping
diffeomorphisms to the tangent space of a Hilbert sphere, a vector space whose
elements can be expressed using an orthogonal basis. This framework is first
applied to shape-constrained univariate, unconditional pdf estimation and then
extended to conditional pdf estimation. We derive asymptotic convergence rates
of the estimator and demonstrate this approach using a synthetic dataset
involving speed distribution for different traffic flow on Californian
driveways.
