Nonquadratic estimators of a quadratic functional. (English) Zbl 1085.62055

Cai, T. Tony; Low, Mark G.

Ann. Stat. 33, No. 6, 2930-2956 (2005).

Summary: 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.

MSC:

62G99	Nonparametric inference
62M99	Inference from stochastic processes
46N30	Applications of functional analysis in probability theory and statistics
62F12	Asymptotic properties of parametric estimators
62G07	Density estimation

Keywords:

Besov balls; Gaussian sequence model; information bound; minimax estimation; quadratic estimators

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