Wavelets, approximation, and statistical applications

By Wolfgang Heardle et al.. 2013

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On the optimal estimation of probability measures in weak and strong topologies (2013, positive): “…p , the popular ones being the kernel density estimator and wavelet estimator, which under suitable conditions have been shown to be optimal with respect to the L r loss (1 ≤ r ≤ ∞ ) in the sense of achieving the minimax rate over all estimators for densities in certain classes [8, 22, 37].”
On adaptive inference and confidence bands (2011, methodology): “We shall use basic wavelet theory [6, 14, 27] freely throughout this article, and we shall say that the wavelet basis is S-regular if the corresponding scaling functions φk and wavelets ψk are compactly supported and S-times continuously differentiable on T .”
Degenerate U- and V-statistics under weak dependence : Asymptotic theory and bootstrap consistency (2012, unknown): “For this purpose, we additionally assume (A2∗) (i) θ̂n P−→ θ ∈ ⊆ Rp. (ii) E(h(X∗1, y, θ̂n)|Xn) = 0,∀y ∈ Rd .”
Estimation for L\'{e}vy processes from high frequency data within a long time interval (2011, background): “where Bmn = O(mn) (Bmn is defined in (16)).”
Rates of contraction for posterior distributions in Lr-metrics, 1 ≤ r ≤ ∞ (2011, methodology): “6 in [19], there exist bounded, compactly supported nonnegative functions Φ such that K̃(x, y) ≤ Φ(|x − y|), for all x, y.”
On Bayesian supremum norm contraction rates (2013, methodology): “We refer to the Lecture Notes by Härdle, Kerkyacharian, Picard and Tsybakov [18] for an Introduction to wavelets.”
GLOBAL UNIFORM RISK BOUNDS FOR WAVELET DECONVOLUTION ESTIMATORS (2011, methodology): “We refer to [23, 32] for the basic theory of wavelets that we shall use freely in this article.”
Higher Order Inference On A Treatment Effect Under Low Regularity Conditions (2011, background): “Consider the plug-in estimator ψ(θ) and the one-step estimator ψ(θ) + 1N ∑Ni=1 U1,i θ = 1N ∑Ni=1{Yi −b (Xi)}{Ai −p (Xi)}, whereθ is a rate optimal nonparametric estimator of θ (i.e of FO, (Härdle et al., 1998)).”
Identifying spikes and seasonal components in electricity spot price data: A guide to robust modeling (2013, background): “Note that in terms of smoothing properties wavelet filters show some resemblance to kernel density estimators (KDEs).”