Estimation of convex supports from noisy measurements

Victor Emmanuel Brunel, Jason M. Klusowski, Dana Yang

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

A popular class of problems in statistics deals with estimating the support of a density from n observations drawn at random from a d-dimensional distribution. In the one-dimensional case, if the support is an interval, the problem reduces to estimating its end points. In practice, an experimenter may only have access to a noisy version of the original data. Therefore, a more realistic model allows for the observations to be contaminated with additive noise. In this paper, we consider estimation of convex bodies when the additive noise is distributed according to a multivariate Gaussian (or nearly Gaussian) distribution, even though our techniques could easily be adapted to other noise distributions. Unlike standard methods in deconvolution that are implemented by thresholding a kernel density estimate, our method avoids tuning parameters and Fourier transforms altogether. We show that our estimator, computable in (O(log n))(d−1)/2 time, converges at a rate of Od(log log n/√log n) in Hausdorff distance, in accordance with the polylogarithmic rates encountered in Gaussian deconvolution problems. Part of our analysis also involves the optimality of the proposed estimator. We provide a lower bound for the minimax rate of estimation in Hausdorff distance that is Ωd(1/log2 n).

Original languageEnglish (US)
Pages (from-to)772-793
Number of pages22
JournalBernoulli
Volume27
Issue number2
DOIs
StatePublished - May 2021
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability

Keywords

  • Convex bodies
  • Order statistics
  • Support estimation
  • Support function

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