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## On spatial adaptive estimation of nonparametric regression (1997)

Venue: | Math. Meth. Statistics |

Citations: | 80 - 4 self |

### Citations

983 | Adapting to unknown smoothness via wavelet shrinkage
- DONOHO, JOHNSTONE
- 1995
(Show Context)
Citation Context ...sides this, our adaptive estimate is computationally efficient and demonstrates reasonable practical behavior. 1 Introduction Suppose we are given noisy observations y(x) of a signal – a function f : =-=[0, 1]-=- → R – along the regular grid Γn = {i/n, i = 0, . . . , n}: y(x) = f(x) + ξ(x), x ∈ Γn, (1) where {ξ(x)}x∈Γn is a sequence of independent N (0, 1) random variables defined on the underlying probabilit... |

714 |
Ideal Spatial Adaptation via Wavelet Shrinkage
- Donoho, Johnstone
- 1994
(Show Context)
Citation Context ...tical drawback, the idea turns out to be rather attractive from the practical viewpoint. 7.2 Numerical results As the test signals, we used the functions Blocks, Bumps, HeaviSine and Doppler given in =-=[1, 2]-=- (see formulas therein 1 ). In all experiments, n = 2048. In the first series of the experiments (Figures 1.1 - 1.4), same as in [1, 2], the signals f given by the corresponding formulae were, before ... |

294 | Wavelet Shrinkage: Asymptopia
- Donoho, Johnstone, et al.
- 1993
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Citation Context ...lasses F the actual signal belongs to, and thus meet with severe difficulties when trying to use ”highly specialized” estimates (for detailed discussion of these conceptual issues, see Donoho et. al. =-=[3]-=-). As a result, there is a strong interest in developing adaptive estimates which are optimal in order not over a single class F, but over a family of these classes (the wider is the family, the more ... |

116 |
Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors
- Lepski, Mammen, et al.
- 1997
(Show Context)
Citation Context ... segment. Recently, several spatial adaptive estimates were proposed (the wavelet-based estimators of Donoho et al. [2, 3], and Juditsky [4], adaptive kernel estimates of Lepskii, Mammen and Spokoiny =-=[7]-=-); in the cited papers, the smoothness of the signal is specified as membership in the Besov or Triebel spaces. The goal of this paper is to develop an estimator which is spatial adaptive with respect... |

60 |
A problem of adaptive estimation in Gaussian white noise. Teor. Veroyatnost. i Primenen
- Lepskĭı
- 1990
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Citation Context ...sured in the L2 scale), at least in the case of smooth periodic signals (the periodicity assumption was later eliminated). In contrast to this, for the problem of estimating at a given point, Lepskii =-=[5]-=-, along with developing kernel estimator with adaptively adjusted smoothing parameter, has shown that it is impossible to adapt a “point” estimator to the (unknown) smoothness of the signal, and here ... |

47 |
A Learning Algorithm for Nonparametric Filtering
- Efroimovich, Pinsker
- 1984
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Citation Context ...mily of these classes (the wider is the family, the more attractive, from the practical viewpoint, is the estimate). The breakthrough in constructing adaptive estimates is due to Pinsker, Efroimovich =-=[9]-=- who developed an optimal in order estimator (even with C(n) = 1 + o(1) as n → ∞), with respect to the accuracy measure µ = (2, [0, 1], 2), over all ellipsoids ∞� ∞� F = {f(·) = fiei(·) : a i=1 i=1 2 ... |

43 |
Asymptotically minimax adaptive estimation. I. Upper bounds. Optimally adaptive estimates. Teor. Veroyatnost. i Primenen
- Lepskĭı
- 1991
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Citation Context ...known) smoothness of the signal, and here only ”ln n-spoiled”, as compared to the case of known smoothness, risks can be achieved; for general theory of adaptive nonparametric estimation, see Lepskii =-=[6]-=-. 2sFrom practical viewpoint, an extremely important property of a ”good” estimator is its spatial adaptivity, the property as follows: if there exists a segment ¯ ∆ ⊂ [0, 1] which covers the point x0... |

31 | Wavelet estimators: adapting to unknown smoothness
- Juditsky
(Show Context)
Citation Context ...vance ¯ ∆ and the related parameters of smoothness of f on this segment. Recently, several spatial adaptive estimates were proposed (the wavelet-based estimators of Donoho et al. [2, 3], and Juditsky =-=[4]-=-, adaptive kernel estimates of Lepskii, Mammen and Spokoiny [7]); in the cited papers, the smoothness of the signal is specified as membership in the Besov or Triebel spaces. The goal of this paper is... |

21 |
Nonparametric Estimation of Smooth Regression Function
- Nemirovskii
- 1985
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Citation Context ...o � � ln n • φ(n) = n l−1/p+1/q 2l+1−2/p , if q ≥ p(2l + 1) (case I, “strong” accuracy measure); • φ(n) = (ln n) l 2l+1 n −l/(2l+1) , if q ≤ p(2l + 1) (case II, “weak” accuracy measure). It is known (=-=[8]-=-) that in Case I the optimal risk R ∗ behaves itself exactly as φ(·), so that in this case our adaptive estimate is optimal in order. It is well-known that in the remaining case the optimal risk R ∗ i... |