Level set and density estimation on manifolds

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 subset of R-d, from an iid sample of X. To do that we introduce a kernel-based estimator (f) over cap (n,h), which is a slightly modified version of the one proposed in Rodriguez-Casal and Saavedra-Nieves (2014) 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) over capn,L-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) over capn,L-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. (C) 2021 Elsevier Inc. All rights reserved.