Level set and density estimation on manifolds

Abstract

We tackle the problem of the estimation of the level sets L_f({\lambda}) of the density f of a random vector X supported on a smooth manifold M\subsetR^d , from an iid sample of X. To do that we introduce a kernel-based estimator f^n,h , which is a slightly modified version of the one proposed in [45], and proves its a.s. uniform convergence to f . Then, we propose two estimators of L f ({\lambda}), the first one is a plug-in: L f^n,h ({\lambda}), which is proven to be a.s. consistent in Hausdorff distance and distance in measure, if L f({\lambda}) does not meet the boundary of M . While the second one assumes that L f({\lambda}) is r-convex, and is estimated by means of the r-convex hull of L f^n,h({\lambda}). The performance of our proposal is illustrated through some simulated examples. In a real data example we analyze the intensity and direction of strong and moderate winds.