Shape-Constrained and Unconstrained Density Estimation Using Geometric Exploration

The problem of nonparametrically estimating probability density functions (pdfs) from observed data requires posing and solving optimization problems on the space of pdfs. We take a geometric approach and explore this space for optimization using actions of a time-warping group. One action, termed area preserving, is transitive and is applicable to the case of unconstrained density estimation. In this case, we take a two-step approach that involves obtaining any initial estimate of the pdf and then transforming it via this warping action to reach the final estimate by maximizing the log-likelihood function. Another action, termed mode-preserving, is useful in situations where the pdf is constrained in shape, i.e. the number of its modes is known. As earlier, we initialize the estimation with an arbitrary element of the correct shape class, and then search over all time warpings to reach the optimal pdf within that shape class. Optimization over warping functions is performed numerically using the geometry of the group of warping functions. These methods are illustrated using a number of simulated examples.