Abstract
The global lower bound for the minimax risk proposed in Part I [12] is applied to the pointwise estimation of functions in the white Gaussian noise, under the squared losses. Some general ellipsoidal and cuboidal functional classes are discussed, including classes of entire functions of exponential type, Paley-Wiener classes of analytic functions, Sobolev classes and their modifications. Based on the proposed risk bounds, a numerical comparison of the minimax risks and the linear minimax risks is made. A nonasymptotic comparison of different types of functional classes is facilitated by their respective embeddings provided the classes are properly calibrated. This discussion demonstrates that the commonly perceived notion of a close connection between the smoothness of an unknown function and the accuracy of estimation can be misleading in a nonasymptotic setting. In particular, the notion of optimal rates of convergence, which has dominated nonparametric statistics for the last three decades, may no longer be productive.
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Levit, B. Minimax revisited. II. Math. Meth. Stat. 19, 299–326 (2010). https://doi.org/10.3103/S1066530710040010
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DOI: https://doi.org/10.3103/S1066530710040010
Keywords
- white Gaussian noise
- linear functionals
- ellipsoidal classes
- cuboidal classes
- Paley-Wiener classes
- Sobolev classes