Uniform limit theorems for wavelet density estimators

By Evarist Gin'e, Richard Nickl. 2008

84 citations recorded by the source corpus.

Let pn(y) = Σ k Φ (y-k) + Σ jn-1 l=0 Σ k lk 2l/2 ψ(2 l y-k) be the linear wavelet density estimator, where φ, ψ are a father and a mother wavelet (with compact support ? ??? are the empirical wavelet coefficients based on an i.i.d. sample of random variables distributed according to a density p 0 on R, and j n ∈ Z, j n ∞. Several uniform limit theorems are proved: First, the almost sure rate of convergence of sup y∈R | Pn (y) - Ep n (y)| is obtained, and a law of the logarithm for a suitably scaled version of this quantity is established. This implies that sup y∈R |p n (y) - po (y)| Attains the optimal almost sure rate of convergence for estimating p 0 , if j n is suitably chosen. Second, a uniform central limit theorem as well as strong invariance principles for the distribution function of p n , that is, for the stochastic processes n(F W n (s) - F (s)) n∫ s - ∞ (pn - p 0 ), s ∈ R, are proved; and more generally, uniform central limit theorems for the processes n∫(p n - P 0 )?, f ∈ F, for other Donsker classes F of interest are considered. As a statistical application, it is shown that essentially the same limit theorems can be obtained for the hard thresholding wavelet estimator introduced by Donoho et al. [Ann. Statist. 24 (1996) 508-539].

Subfields: Computer Vision and Pattern Recognition, Finance, Statistics and Probability

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Citations in the corpus

On the optimal estimation of probability measures in weak and strong topologies (2013, background): “…and the corresponding uniform central limit theorem (UCLT) hold if P ⋆n − P n F = o P ( n − 1 / 2 ), i.e., sup Several recent works [6, 26, 16, 17, 18, 19] have shown that many popular density estimators on X = R , such as maximum likelihood estimator, kernel density estimator and wavelet…”
Nonparametric Bernstein–von Mises theorems in Gaussian white noise (2012, background): “Standard frequentist estimators such as kernel, wavelet and nonparametric maximum likelihood estimators satisfy this property; in fact, one can even prove a corresponding uniform central limit theorem in l∞(H) for such estimators; see Kiefer and Wolfowitz [24], Nickl [30], Giné and Nickl [21, 22].”
CONFIDENCE BANDS IN DENSITY ESTIMATION (2010, methodology): “Condition (K5) follows from Lemma 2 in Giné and Nickl (2009b).”
On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures (2013, background): “≤ C‖h‖M0, where we have used supt∈[0,1] ∑ k |〈1[0,t],ψlk〉| ≤ c2−l/2, shown, for example, as in the proof of Lemma 3 in [16].”
A Donsker Theorem for Lévy Measures (2012, background): “The following general purpose result follows from the proof of Theorem 3 in Giné and Nickl (2008), which builds on fundamental ideas in the classical paper Giné and Zinn (1984), and can be applied to the unbounded processes relevant here. For a given class of measurable functions F we write F ′ δ = ”
Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections (2008, methodology): “Now, using these facts, one can furthermore show that the wavelet series (3) converges in Lp(R), p ∞, for f ∈ Lp(R) and we also note that if p0 is a uniformly continuous density, then its wavelet series converges uniformly.”
Adaptive Bernstein–von Mises theorems in Gaussian white noise (2014, background): “See Giné and Nickl [21] and also Bull [6] for further discussion about this condition.”
GLOBAL UNIFORM RISK BOUNDS FOR WAVELET DECONVOLUTION ESTIMATORS (2011, methodology): “This will complete the proof of the lemma by using Lemma 1 in [19], which states that the set of dilates and translates of a fixed function h of bounded p-variation, 1 ≤ p < ∞, is of VC-type with constants A,v depending only on p and the pvariation norm of h.”
Adaptive density estimation: a curse of support? ✩ (2009, unknown): “Note that the case p ′ = ∞ is very di ﬀ erent in nature (see Gin´e and Nickl (2009) and Gin´e and Nickl (2010)).”
Rates of contraction for posterior distributions in Lr-metrics, 1 ≤ r ≤ ∞ (2011, methodology): “It is convenient to observe that the degree of concentration of a kernel-type density estimator around its expectation in Lr depends on r, as can already be seen from comparing the known cases r = 1,∞ in [14, 16] for kernel estimators and [17] for wavelets.”
An exponential inequality for the distribution function of the kernel density estimator, with applications to adaptive estimation (2009, background): “In the recent articles [2,17,18,29], it was shown that several nonparametric density estimators (such as maximum likelihood, wavelet or kernel estimators) are within a ‖ · ‖F -ball of probabilistic size o(1/√n) around the empirical measure, where ‖μ‖F = sup f ∈F | ∫ f dμ| is the usual supnorm of a m”
On adaptive inference and confidence bands (2011, background): “About question (ii): if the radius B of the Hölder balls involved is known, then one can combine a natural testing approach with recent results in [10, 11, 13] to prove the existence of adaptive nonparametric confidence bands over a scale of Hölder classes indexed by a grid of smoothness parameters ”
Gaussian approximation of suprema of empirical processes (2012, background): “We note that [46] and especially [27, 28] give general sufficient conditions under which K is VC type.”
Plugin estimation of smooth optimal transport maps (2021, background): “Part (iii) can be proven using a similar strategy, or can easily be obtained from adaptations of existing L∞ concentration inequalities for wavelet density estimators over Rd (Masry, 1997; Giné and Nickl, 2009; Guo and Kou, 2019).”
Anti-concentration and honest, adaptive confidence bands (2013, background): “., are nested, and ∪j≥0Vj is dense in L2(R). Suppose that φ is compactly supported, bounded, and of bounded p-variation for some p≥ 1. For example, all Daubechies’ wavelets have these properties (see [12], p. 1613). Suppose also that there exists a constant c>0 such that for all x∈ Rd, X k∈Zd Y”