Multiscale methods for shape constraints in deconvolution: Confidence statements for qualitative features

Schmidt-Hieber, Johannes; Munk, Axel; Dümbgen, Lutz

doi:10.1214/13-aos1089

Cited by 48 publications

(61 citation statements)

References 44 publications

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“…A similar situation arises in standard deconvolution problems, see the recent paper [24]. In the spirit of the theory of pseudo-differential operators this is established by differentiating in the spectral domain, see (4.10) below for details,…”

Section: The Regularity Conditionsmentioning

confidence: 83%

A Donsker theorem for Lévy measures

Nickl

Reiß

2012

Journal of Functional Analysis

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Given n equidistant realisations of a Lévy process (L t , t 0), a natural estimatorN n for the distribution function N of the Lévy measure is constructed. Under a polynomial decay restriction on the characteristic function ϕ, a Donsker-type theorem is proved, that is, a functional central limit theorem for the process √ n(N n − N) in the space of bounded functions away from zero. The limit distribution is a generalised Brownian bridge process with bounded and continuous sample paths whose covariance structure depends on the Fourier-integral operator F −1 [1/ϕ(−•)]. The class of Lévy processes covered includes several relevant examples such as compound Poisson, Gamma and self-decomposable processes. Main ideas in the proof include establishing pseudo-locality of the Fourier-integral operator and recent techniques from smoothed empirical processes.

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Section: The Regularity Conditionsmentioning

confidence: 83%

A Donsker theorem for Lévy measures

Nickl

Reiß

2012

Journal of Functional Analysis

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“…In the context of density estimation, multiscale tests have been investigated in Dümbgen and Walther (), Rufibach and Walther (), Schmidt‐Hieber et al . () and Eckle et al . () among others.…”

Section: Introductionmentioning

confidence: 87%

“…More recently, Proksch et al (2018) have constructed multiscale tests for inverse regression models. In the context of density estimation, multiscale tests have been investigated in Dümbgen and Walther (2008), Rufibach and Walther (2010), Schmidt-Hieber et al (2013) and Eckle et al (2017) among others.…”

Section: Introductionmentioning

confidence: 99%

Multiscale Inference and Long-Run Variance Estimation in Non-Parametric Regression with Time Series Errors

Khismatullina

Vogt

2019

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary We develop new multiscale methods to test qualitative hypotheses about the function m in the non‐parametric regression model Yt,T=m(t/T)+ɛt with time series errors ɛt. In time series applications, m represents a non‐parametric time trend. Practitioners are often interested in whether the trend m has certain shape properties. For example, they would like to know whether m is constant or whether it is increasing or decreasing in certain time intervals. Our multiscale methods enable us to test for such shape properties of the trend m. To perform the methods, we require an estimator of the long‐run error variance σ2=normalΣl=−∞∞covfalse(ε0,εlfalse). We propose a new difference‐based estimator of σ2 for the case that {ɛt} belongs to the class of auto‐regressive AR(∞) processes. In the technical part of the paper, we derive asymptotic theory for the proposed multiscale test and the estimator of the long‐run error variance. The theory is complemented by a simulation study and an empirical application to climate data.

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“…Another way is to use a bound on the (1 ↵)-quantile of f W n using sharp deviation inequalities available to Gaussian processes, which leads to analytic construction of confidence bands; see, for example, [14] for this approach. In some applications, the distribution of the approximating Gaussian process is completely known, and in that case the distribution of f W n can be simulated via a direct Monte Carlo method; see [52] for such examples. Finally, we mention that there are alternative, yet more conservative, approaches on construction of confidence bands based on non-asymptotic concentration inequalities (and not on Gaussian approximation); see [40] and [33].…”

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confidence: 99%

Gaussian approximation of suprema of empirical processes

Chernozhukov¹,

Chetverikov²,

Kato³

2016

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This paper develops a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of Gaussian processes, without taking the route of approximating whole empirical processes in the sup-norm. We prove an abstract approximation theorem applicable to a wide variety of statistical problems, such as construction of uniform confidence bands for functions. Notably, the bound in the main approximation theorem is nonasymptotic and the theorem does not require uniform boundedness of the class of functions. The proof of the approximation theorem builds on a new coupling inequality for maxima of sums of random vectors, the proof of which depends on an e↵ective use of Stein's method for normal approximation, and some new empirical process techniques. We study applications of this approximation theorem to local and series empirical processes arising in nonparametric estimation via kernel and series methods, where the classes of functions change with the sample size and are non-Donsker. Importantly, our new technique is able to prove the Gaussian approximation for the supremum type statistics under weak regularity conditions, especially concerning the bandwidth and the number of series functions, in those examples.

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Multiscale methods for shape constraints in deconvolution: Confidence statements for qualitative features

Cited by 48 publications

References 44 publications

A Donsker theorem for Lévy measures

A Donsker theorem for Lévy measures

Multiscale Inference and Long-Run Variance Estimation in Non-Parametric Regression with Time Series Errors

Gaussian approximation of suprema of empirical processes

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