Adaptive function estimation in nonparametric regression with one-sided errors

Abstract

We consider the model of nonregular nonparametric regression where smoothness constraints are imposed on the regression function f and the regression errors are assumed to decay with some sharpness level at their endpoints. The aim of this paper is to construct an adaptive estimator for the regression function f . In contrast to the standard model where local averaging is fruitful, the nonregular conditions require a substantial different treatment based on local extreme values. We study this model under the realistic setting in which both the smoothness degree β > 0 and the sharpness degree a ∈ (0, ∞) are unknown in advance. We construct adaptation procedures applying a nested version of Lepski's method and the negative Hill estimator which show no loss in the convergence rates with respect to the general L q -risk and a logarithmic loss with respect to the pointwise risk. Optimality of these rates is proved for a ∈ (0, ∞). Some numerical simulations and an application to real data are provided.

npbr: A Package for Nonparametric Boundary Regression in R

“…These annual records, depicted in Figure 1(a), display some interesting features. Following Jirak, Meister, and Reiss (2014), the lower boundary can be interpreted as the best possible time for a given year. This boundary steadily decreases from 1970 until around the year 2000, followed by a sudden increase.…”

Laguerre deconvolution with unknown matrix operator

“…More broadly the problem of nonnegative variables appears in actuarial or insurance models. Recently, in a financial context, some papers as Jirak et al (2014) or Reiß and Selk (2015) have addressed the problem of one-sided errors. The first authors are interested in the optimal adaptive estimation in nonparametric regression when the errors are not assumed to be centered anymore, and typically with Exponential density.…”

Semi-parametric transformation boundary regression models

“…In contrast in the paper at hand we consider boundary regression models. Such nonparametric regression models with one-sided errors have been considered, among others, by Hall and Van Keilegom (2009), Meister and Reiß (2013), Jirak, Meister and Reiß (2014) and Drees, Neumeyer and Selk (2018). Relatedly, estimation of support boundaries have been considered, for instance, by Härdle, Park and Tsybakov (1995), Hall, Park and Stern (1998), Girard and Jacob (2008) and Daouia, Noh and Park (2016).…”

“…The equidistant fixed design -as well as its natural generalization to deterministic covariates -is often used in real-life applications when time is involved in the data set. This is the case for instance in Jirak et al (2014) where the authors studied the monthly sunspot observations and the annual best running times of 1500 meters. Besides, deterministic design is met accross a number of papers in regression models, see for instance Brown and Low (1996), Meister and Reiß (2013) and the references within.…”