Level set and density estimation on manifolds

Abstract

We tackle the problem of the estimation of the level sets Lf (λ) of the density f of a random vector X supported on a smooth manifold M ⊂ Rd, from an iid sample of X. To do that we introduce a kernel-based estimator ˆfn,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 Lf (λ), the first one is a plug-in: L ˆfn,h (λ), which is proven to be a.s. consistent in Hausdorff distance and distance in measure, if Lf (λ) does not meet the boundary of M . While the second one assumes that Lf (λ) is r-convex, and is estimated by means of the r-convex hull of L ˆfn,h (λ). 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.