Minimax quadratic estimation of a quadratic functional (original) (raw)

We study the problem of estimating the integrated squared derivative of a periodic function, when the observations are incomplete and noisy. We derive a simple quadratic estimate which is asymptotically minimax among quadratic estimates. Our estimate is also rateoptimal among all measurable estimates. Key Words and Phrases. Moduilus of Continuity. Optimal Rates of Convergence. Ellipsoids. tlypertectangles. Estimation of Linear Functionals. Acknowledgements. This work was partially siupported by NSF grants DMS 84-57153 and D)MS 88-10192. The authors wolild like to thank Jianqing Fan for helpful conversations.