The minimax theory for estimating linear functionals is extended to the case of a finite union of convex parameter spaces. Upper and lower bounds for the minimax risk can still be described in terms of a modulus of continuity. However in contrast to the theory for convex parameter spaces rate optimal procedures are often required to be nonlinear. A construction of such nonlinear procedures is given. The results developed in this paper have important applications to the theory of adaptation.
"Minimax estimation of linear functionals over nonconvex parameter spaces." Ann. Statist. 32 (2) 552 - 576, April 2004. https://doi.org/10.1214/009053604000000094