Adaptive estimation of planar convex sets

Understand

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.