Non-Quadratic Estimators of A Quadratic Functional
Tony Cai and >Mark Low
Estimation of a quadratic functional over parameter spaces that are not quadratically convex is considered. It is shown, in contrast to the theory for quadratically convex parameter spaces, that optimal quadratic rules are often
rate suboptimal. In such cases minimax rate optimal procedures are constructed based on local thresholding.
These non-quadratic procedures are sometimes fully efficient even when optimal quadratic rules have slow rates of convergence. Moreover it is shown that when estimating a quadratic functional non-quadratic procedures may exhibit different elbow phenomenon than quadratic procedures.