On Modulus of Continuity And Adaptability in Nonparametric Functional Estimation

We study adaptive estimation of linear functionals over a collection of finitely many parameter spaces.A between class modulus of continuity, a geometric quantity, is introduced and is shown to be instrumental in characterizing the degree of adaptability over two parameter spaces in the same way that the usual modulus of continuity captures the minimax difficulty of estimation over a single parameter space. The between class modulus of continuity is used to describe when full adaptation is possible. A sharp rate optimal lower bound on the cost of adaptation is derived. An ordered modulus of continuity is used to construct a procedure which which is within a constant factor of attaining these bounds. The results are complemented by several illustrative examples.