Tony Cai, University of Pennsylvania

"Adaptive estimation of a planar convex set"

In this talk, I will present some recent work on adaptive estimation of an unknown planar compact, convex set from noisy measurements of its support function. Both the problem of estimating the support function at a point and that of estimating the whole convex set are considered. For pointwise estimation, we study the problem in a general non-asymptotic framework, which evaluates the performance of a procedure at each individual set, instead of the worst case performance over a large parameter space as in the conventional minimax theory. A data-driven adaptive estimator is proposed and is shown to be optimally adaptive to every compact, convex set. For estimating the whole convex set, we propose estimators that are shown to adaptively achieve the optimal rate of convergence.