Consistencies and rates of convergence of jump-penalized least squares estimators

By Leif Boysen, A. Kempe, Volkmar Liebscher, Axel Munk, Olaf Wittich. 2009

187 citations recorded by the source corpus.

We study the asymptotics for jump-penalized least squares regression aiming at approximating a regression function by piecewise constant functions. Besides conventional consistency and convergence rates of the estimates in L 2 ([0,1)) our results cover other metrics like Skorokhod metric on the space of cadlag functions and uniform metrics on C([0, 1]). We will show that these estimators are in an adaptive sense rate optimal over certain classes of "approximation spaces." Special cases are the class of functions of bounded variation (piecewise) Holder continuous functions of order 0 < α ≤ 1 and the class of step functions with a finite but arbitrary number of jumps. In the latter setting, we will also deduce the rates known from change-point analysis for detecting the jumps. Finally, the issue of fully automatic selection of the smoothing parameter is addressed.

Subfields: Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a7fdcb728960dd2ac30683837510de2128107f13

Citations in the corpus

• Heterogeneous change point inference (2015, unknown): “ted models are vast, see for instance (Yao, 1988; Donoho and Johnstone, 1994; Csorgo and Horvath, 1997; Bai and Perron, 1998; Braun et al., 2000; Birge and Massart, 2001; Kolaczyk and Nowak, 2005; Boysen et al., 2009; Harchaoui and Levy-Leduc, 2010; Jeng et al., 2010; Killick et al., 2012; Rigollet ”
• Wild binary segmentation for multiple change-point detection (2014, unknown): “Boysen et al. (2009) use the least-squares criterion with a linear penalty on the number of change-points and, under the assumption of a finite but arbitrary N , show various theoretical results including analogues of those of Yao and Au (1989).”
• Multiple Change-Point Estimation With a Total Variation Penalty (2010, unknown): “T n , K = {ˆ t 1 ,... , ˆ t K } of K estimated change points and the set of the true change points T n = { t 1 ,... , t K } by using as in Boysen et al. (2009) Note that we recover the Hausdorff distance between the sets A and B with Obviously, when K = K (cid”
• Univariate mean change point detection: Penalization, CUSUM and optimality (2018, unknown): “• There exist several results and algorithms for change point detection using ℓ 0 penalization, including Liebscher and Winkler (1999), Friedrich et al. (2008), Boysen et al. (2009) and Killick et al. (2012).”
• Group LASSO for Structural Break Time Series (2014, unknown): “To evaluate the consistency properties of the change-points, define the Hausdorff distance between two sets A and B as in Boysen et al. (2009) by dH (A,B) = max b∈B min a∈A |b − a|, where dH (A,∅) = dH (∅, B) = 1, and ∅ is the empty set.”
• A review of change point detection methods (2018, unknown): “Distances |t̂k − tk| between true change point indexes and their estimated counterparts do not converge to 0, even for simple models [6, 7, 17, 26, 30, 129].”
• A robust approach for estimating change-points in the mean of an AR(1) process (2014, unknown): “In order to assess the performance of the estimation of the change-point locations, we computed the Hausdorff distance defined in the segmentation framework as follows, see [7,16]:”
• Convergence rates of least squares regression estimators with heavy-tailed errors (2017, unknown): “From an estimation viewpoint, [11] proposed an 0-type penalized LSE for estimating regression functions in Fk , where a (nearly) parametric rate is obtained under a sub-Gaussian condition on the errors.”
• Selective review of offline change point detection methods (2020, unknown): “Hausdorff 206 The Hausdorff metric, denoted ∆HA, measures the robustness of detection 207 methods [47, 48].”