Minimax revisited. II |
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Authors: | B Levit |
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Institution: | 1.Dept. of Math. and Statist.,Queen’s University,Kingston,Canada |
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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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Keywords: | |
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