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 language | English (US) |
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Pages (from-to) | 772-793 |
Number of pages | 22 |
Journal | Bernoulli |
Volume | 27 |
Issue number | 2 |
DOIs | |
State | Published - May 2021 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- Statistics and Probability
Keywords
- Convex bodies
- Order statistics
- Support estimation
- Support function