  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 nonquadratic 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 nonquadratic
procedures may exhibit different elbow phenomena than quadratic procedures.
