Level set and density estimation on manifolds
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Given an iid sample of a distribution supported on a smooth manifold M^d , which is assumed to be absolutely continuous w.r.t the Hausdorff measure inherited from the ambient space, we tackle the problem of the estimation of the level sets of the density f . A consistent estimator in both Hausdorff distance and distance in measure is proposed. The estimator is the level set of the kernel-based estimator of the density f . We prove that the kernel-based density estimator converges uniformly to the unknown density f , the consistency of the level set and the consistency of the boundary of the level set estimator. The performance of our proposal is illustrated through some simulated examples.
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