On consistency of redescending M-kernel smoothers

Abstract

Robust regression, Nonparametric regression, M-estimation, Jump preserving M-kernel estimation, Consistency, 62 G07, 62 G08, 62 G35,

Simple consistent cluster methods based on redescending M-estimators with an application to edge identification in images

“…This is an approach used also by Chu et al [5] for constructing corner preserving M-smoothers for image reconstruction. The consistency of these M-smoothers even at jumps was shown by Hillebrand and Mu¨ller [12]. A similar proof can be used here to prove the consistency of the set M N ðy N Þ: In Section 5 it is shown that the set M N ðy N Þ is a consistent estimator for the set M: For that we need not only pointwise convergence of H N ðm; y N Þ to hðmÞ but also uniform convergence which can be achieved by intersecting M N ðy N Þ with a compact subset of R k : Appropriate compact subsets are given by…”

“…Condition (10) is essential but often overlooked in similar approaches (see [12]). Contrary to the other conditions, it is in general not easy to verify.…”

Design of kernel M-smoothers for spatial data

“…(13)(a) shows that the MSE does not vanish asymptotically and, therefore, the M-smoother is not consistent at the jump-point x 0 . However, the formula of π shows that if f (ε) has a bounded support, with range less than δ, then the consistency may exist (see also Hillebrand and Müller [11]). …”

Robust nonparametric estimation of the intensity function of point data

“…Their main field of application was regression models contaminated by outliers. Bivariate M-smoothers have proved effective in preserving edges in digital image denoising (see Chu et al 1998;Rue et al 2002;Hillebrand and Müller 2006); this approach can potentially be applied to smooth point data and their histograms. The robust (M-type) local polynomial regression (LPR) estimator of the intensity model (2) can be defined aŝ…”

“…These estimators have been successfully applied as edge-preserving filters to denoise digital images (e.g. Chu et al 1998;Rue et al 2002;Hillebrand and Müller 2006). Common smoothers reduce the noise by local averaging of the pixel luminance; however, this transformation also blurs the edges which separate homogeneous zones.…”