A geometric framework for density estimation and conditional density estimation

We introduce a geometrically intuitive procedure to obtain an estimator for a probability density function in the absence or presence of predictors. The estimation procedure is based on starting with an initial estimate of the density shape and then transforming it via a warping function to obtain the final estimate. The idea is to design the initial estimate to be computationally fast, albeit suboptimal, and then use the warping to create a flexible class of density functions, resulting in substantially improved estimation. The second step is accomplished by mapping warping functions to the tangent space of a Hilbert sphere, a vector space whose elements can be expressed using an orthogonal basis. Using a truncated basis expansion, we use constrained optimization to estimate optimal warping and, thus, the optimal density estimate. This framework is introduced in a univariate density estimation setup and then extended to conditional density estimation and multivariate density estimation. The approach provides a good balance of excellent practical performance and efficiency, avoiding many of the computational issues associated with conventional methods, which is illustrated using simulated datasets.