Abstract
Estimation of a quadratic functional of a function observed in the Gaussian white noise model is considered. A data-dependent method for choosing the amount of smoothing is given. The method is based on comparing certain quadratic estimators with each other. It is shown that the method is asymptotically sharp or nearly sharp adaptive simultaneously for the “regular” and “irregular” region. We consider l p bodies and construct bounds for the risk of the estimator which show that for p=4 the estimator is exactly optimal and for example when p ∈[3,100], then the upper bound is at most 1.055 times larger than the lower bound. We show the connection of the estimator to the theory of optimal recovery. The estimator is a calibration of an estimator which is nearly minimax optimal among quadratic estimators.
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Writing of this article was financed by Deutsche Forschungsgemeinschaft under project MA1026/6-2, CIES, France, and Jenny and AnttiWihuri Foundation.
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Klemelä, J. Sharp adaptive estimation of quadratic functionals. Probab. Theory Relat. Fields 134, 539–564 (2006). https://doi.org/10.1007/s00440-005-0447-2
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DOI: https://doi.org/10.1007/s00440-005-0447-2
Mathematics Subject Classification (2000)
- Primary 62G07
- Secondary 62G20
Key words or phrases
- Adaptive curve estimation
- Exact constants in nonparametric smoothing
- Minimax risk
- Optimal recovery
- Quadratic functionals
Ser. A 50, 381–393 (1988)