Non-Quadratic Estimators of A Quadratic Functional
The Annals of Statistics 33, 2930-2956, (2005).

Tony Cai and Mark Low

---

• Abstract: 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.

• Paper: pdf file.

• Other related paper:

• Cai, T. & Low, M. (2006).
  Optimal adaptive estimation of a quadratic functional.
  The Annals of Statistics 34, 2298-2325.

  Cai, T. & Low, M. (2006).
  Adaptive confidence balls.
  The Annals of Statistics 34, 202-228.