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Locally Adaptive Density Estimation on the Unit Sphere Using Needlets

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Abstract

The problem of estimating a probability density function f on the d−1-dimensional unit sphere S d−1 from directional data using the needlet frame is considered. It is shown that the decay of needlet coefficients supported near a point x∈S d−1 of a function f:S d−1→ℝ depends only on local Hölder continuity properties of f at x. This is then used to show that the thresholded needlet estimator introduced in Baldi, Kerkyacharian, Marinucci, and Picard (Ann. Stat. 39, 3362–3395, 2009) adapts to the local regularity properties of f. Moreover, an adaptive confidence interval for f based on the thresholded needlet estimator is proposed, which is asymptotically honest over suitable classes of locally Hölderian densities.

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Authors and Affiliations

  1. Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, CB3 0WB, Cambridge, UK

    A. Kueh

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  1. A. Kueh
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Correspondence to A. Kueh.

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Communicated by G. Kerkyacharian.

This forms part of the author’s PHD thesis written under the supervision of Richard Nickl, whose expertise, understanding, and patience helped shape this paper. I also have much gratitude to the referees who wrote such extensive reports helping me to improve this paper. All mistakes remain mine.

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Kueh, A. Locally Adaptive Density Estimation on the Unit Sphere Using Needlets. Constr Approx 36, 433–458 (2012). https://doi.org/10.1007/s00365-012-9170-2

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  • DOI: https://doi.org/10.1007/s00365-012-9170-2

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