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Level set and density estimation on manifolds

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  • Cholaquidis, Alejandro
  • Fraiman, Ricardo
  • Moreno, Leonardo

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 fˆn,h, which is a slightly modified version of the one proposed in Rodríguez-Casal and Saavedra-Nieves (2014) and proves its a.s. uniform convergence to f. Then, we propose two estimators of Lf(λ), the first one is a plug-in: Lfˆn,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 Lfˆn,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.

Suggested Citation

  • Cholaquidis, Alejandro & Fraiman, Ricardo & Moreno, Leonardo, 2022. "Level set and density estimation on manifolds," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    • Handle: RePEc:eee:jmvana:v:189:y:2022:i:c:s0047259x21001925
    DOI: 10.1016/j.jmva.2021.104925
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    References listed on IDEAS

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    5. Baíllo, Amparo, 2003. "Total error in a plug-in estimator of level sets," Statistics & Probability Letters, Elsevier, vol. 65(4), pages 411-417, December.
    6. Ilya S. Molchanov, 1998. "A Limit Theorem for Solutions of Inequalities," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 235-242, March.
    7. Aneiros, Germán & Cao, Ricardo & Fraiman, Ricardo & Genest, Christian & Vieu, Philippe, 2019. "Recent advances in functional data analysis and high-dimensional statistics," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 3-9.
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