2015/08/15 by Tommaso Cai, Tony Cai, Cai, Tony +4 · 1 citation
Engineering · Mathematics · #Control Systems and Identification #FOS: Mathematics #Statistical Methods and Inference #Statistics Theory (math.ST) #Water resources management and optimization #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.1508.03744
arxiv created 2015/08/15 · openalex publication_date 2015/08/15 · arxiv updated 2015/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we consider adaptive estimation of an unknown planar compact, convex set from noisy measurements of its support function on a uniform grid. Both the problem of estimating the support function at a point and that of estimating the convex set are studied. Data-driven adaptive estimators are proposed and their optimality properties are established. For pointwise estimation, it is shown that the estimator optimally adapts to every compact, convex set instead of a collection of large parameter spaces as in the conventional minimax theory of nonparametric estimation. For set estimation, the estimators adaptively achieve the optimal rate of convergence. In both these problems, our analysis makes no smoothness assumptions on the unknown sets.