Estimation of convex supports from noisy measurements

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 ${(\mathit{O}(log\mathit{n}))}^{(\mathit{d}-1)/2}$ time, converges at a rate of ${\mathit{O}}_{\mathit{d}}(loglog\mathit{n}/\sqrt{log\mathit{n}})$ 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 ${\mathrm{\Omega }}_{\mathit{d}}(1/{log}^2\mathit{n})$.