Wavelets, Approximation, and Statistical Applications

Härdle, Wolfgang Karl; Kerkyacharian, Gérard; Tsybakov, Alexandre B.; Picard, Dominique

doi:10.1007/978-1-4612-2222-4

Cited by 586 publications

(580 citation statements)

References 111 publications

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“…It follows from Lemma 8.6 in [23] that the series k∈Z φ(· − k) 2 converges uniformly, and hence k∈Z φ(· − k) 2 is continuous (the functions k:|k|≤m φ(· − k) 2 are continuous for all m by our assumptions on the regularity of the farther wavelet). Further, it follows from Corollary 8.1 in [23] that k∈Z φ(· − k) is identically equal to some non-zero constant, so that…”

Section: Supplemental Materialsmentioning

confidence: 92%

“…in [23]. Therefore, |K l (y, x)| ≤ 2 ld C for all y ∈ X , |E f [K l (X 1 , x)]| ≤ C, and In addition, for convolution kernels, where the equality follows from orthonormality in L 2 (R) of the system {φ(· − k), k ∈ Z} and the inequality follows from Lemma E.1.…”

Section: Supplemental Materialsmentioning

confidence: 97%

“…It is useful to note that the upper bound in (4.2) holds for all f ∈ Σ(t, L) (for sufficiently large C 3 ) under Condition L1; see, for example, Theorem 9.3 in [23]. In addition, [19] showed that under Condition L1, the restriction due to the lower bound in (4.2) is weak in the sense that the set of elements of Σ(t, L) for which the lower bound in (4.2) does not hold is "topologically small".…”

Section: Honest and Adaptive Confidence Bands In Hölder Classesmentioning

confidence: 99%

“…Letf n (·, l) be a generic estimator of f with a smoothing parameter l, say bandwidth or resolution level, where l is chosen from a candidate set L n ; see [23,39,42] for a textbook level introduction to the theory of density estimation. Letl n =l n (X 1 , .…”

Section: Introductionmentioning

confidence: 99%

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Anti-concentration and honest, adaptive confidence bands

Kato¹,

Chetverikov²,

Chernozhukov³

2013

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Abstract. Modern construction of uniform confidence bands for nonparametric densities (and other functions) often relies on the the classical Smirnov-Bickel-Rosenblatt (SBR) condition; see, for example, . This condition requires the existence of a limit distribution of an extreme value type for the supremum of a studentized empirical process (equivalently, for the supremum of a Gaussian process with the same covariance function as that of the studentized empirical process). The principal contribution of this paper is to remove the need for this classical condition. We show that a considerably weaker sufficient condition is derived from an anti-concentration property of the supremum of the approximating Gaussian process, and we derive an inequality leading to such a property for separable Gaussian processes. We refer to the new condition as a generalized SBR condition. Our new result shows that the supremum does not concentrate too fast around any value.We then apply this result to derive a Gaussian multiplier bootstrap procedure for constructing honest confidence bands for nonparametric density estimators (this result can be applied in other nonparametric problems as well). An essential advantage of our approach is that it applies generically even in those cases where the limit distribution of the supremum of the studentized empirical process does not exist (or is unknown). This is of particular importance in problems where resolution levels or other tuning parameters have been chosen in a data-driven fashion, which is needed for adaptive constructions of the confidence bands. Furthermore, our approach is asymptotically honest at a polynomial rate -namely, the error in coverage level converges to zero at a fast, polynomial speed (with respect to the sample size). In sharp contrast, the approach based on extreme value theory is asymptotically honest only at a logarithmic rate -the error converges to zero at a slow, logarithmic speed. Finally, of independent interest is our introduction of a new, practical version of Lepski's method, which computes the optimal, non-conservative resolution levels via a Gaussian multiplier bootstrap method.

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Section: Supplemental Materialsmentioning

confidence: 92%

Section: Supplemental Materialsmentioning

confidence: 97%

Section: Honest and Adaptive Confidence Bands In Hölder Classesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Anti-concentration and honest, adaptive confidence bands

Kato¹,

Chetverikov²,

Chernozhukov³

2013

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“…For a systematic discussion of wavelets and their applications in statistics, see the recent monograph by Härdle, et al (1998). The major advantage of wavelet method is its adaptability to the degree of smoothness of the underlying unknown curve.…”

Section: Introductionmentioning

confidence: 99%

On the minimax optimality of block thresholded wavelet estimators with long memory data

Li¹,

Xiao²

2007

Journal of Statistical Planning and Inference

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We consider the estimation of nonparametric regression function with long memory data and investigate the asymptotic rates of convergence of wavelet estimators based on block thresholding. We show that the estimators achieve optimal minimax convergence rates over a large class of functions that involve many irregularities of a wide variety of types, including chirp and Doppler functions, and jump discontinuities. Therefore, in the presence of long memory noise, wavelet estimators still provide extensive adaptivity to many irregularities of large function classes.

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References

1999

Statistical Modeling by Wavelets

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Wavelets, Approximation, and Statistical Applications

Cited by 586 publications

References 111 publications

Anti-concentration and honest, adaptive confidence bands

Anti-concentration and honest, adaptive confidence bands

On the minimax optimality of block thresholded wavelet estimators with long memory data

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