Abstract: We introduce a novel two-step approach for estimating a probability density function (pdf), given its samples, with the second and important step coming from a geometric formulation. The procedure obtains an arbitrary initial estimate which it transforms using a warping function to reach the final estimate. The initial estimate is intended to be computationally fast, albeit suboptimal; however, but its warping creates a larger, flexible class of density functions, resulting in a substantially improved estimate. The optimal warping is determined by mapping warping functions to the tangent space of a Hilbert sphere, which is a vector space with elements that can be expressed using an orthogonal basis. Using a truncated basis expansion, we estimate the optimal warping under a (penalized) likelihood criterion and obtain the final density estimate. This framework is introduced for univariate unconditional pdf estimations, and then extended to include conditional pdf estimations. The approach avoids many of the computational pitfalls associated with classical conditional-density estimation methods, without sacrificing estimation performance. We derive the asymptotic convergence rates of our density estimator, and demonstrate this approach using synthetic data sets and real data, on the relation between a toxic metabolite on pre-term birth.

Key words and phrases: Conditional density, density estimation, Hilbert sphere, sieve estimation, tangent space, warped density, weighted likelihood maximization.