## Abstract

Let a Hölder continuous function f be observed with noise. In the present paper we study the problem of nonparametric estimation of certain nonsmooth functionals of f, specifically, L_{r} norms ∥f∥_{r} of f. Known from the literature results on functional estimation deal mostly with two extreme cases: estimating a smooth (differentiable in L_{2}) functional or estimating a singular functional like the value of f at certain point or the maximum of f. In the first case, the convergence rate typically is n^{-1/2}, n being the number of observations. In the second case, the rate of convergence coincides with the one of estimating the function f itself in the corresponding norm. We show that the case of estimating ∥f∥_{r} is in some sense intermediate between the above extremes. The optimal rate of convergence is worse than n^{-1/2} but is better than the rate of convergence of nonparametric estimates of f. The results depend on the value of r. For r even integer, the rate occurs to be n^{-β/(2β+1-1/r)} where β is the degree of smoothness. If r is not an even integer, then the nonparametric rate n^{-β/(2β+1)} can be improved, but only by a logarithmic in n factor.

Original language | English |
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Pages (from-to) | 221-253 |

Number of pages | 33 |

Journal | Probability Theory and Related Fields |

Volume | 113 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 1999 |

Externally published | Yes |

## Keywords

- Integral norm
- Non-smooth functional
- Rate of estimation

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