Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators

Abstract

a b s t r a c tWe consider inverse regression models with convolution-type operators which mediate convolution on R d (d ≥ 1) and prove a pointwise central limit theorem for spectral regularisation estimators which can be applied to construct pointwise confidence regions. Here, we cope with the unknown bias of such estimators by undersmoothing. Moreover, we prove consistency of the residual bootstrap in this setting and demonstrate the feasibility of the bootstrap confidence bands at moderate sample sizes in a simulation study.

Confidence Bands in Non-Parametric Errors-In-Variables Regression

“…Some researchers have developed confidence bands in related, but simpler, deconvolution problems; see for example Bissantz et al . (), Bissantz and Birke () and Birke et al . ().…”

“…In a nonparametric context, most of the work so far has concentrated on the development of consistent regression estimators. Some researchers have developed confidence bands in related, but simpler, deconvolution problems; see for example Bissantz et al (2007), Bissantz and Birke (2009) and Birke et al (2010). However, despite their practical importance, to our knowledge confidence bands in errors-in-variables non-parametric regression have largely been ignored so far.…”

Estimation of the Derivatives of a Function in a Convolution Regression Model with Random Design

“…Model (1) can also be viewed as a natural extension of some 1-periodic convolution regression models as those considered by, for example, Cavalier and Tsybakov [1], Pensky and Sapatinas [2], and Loubes and Marteau [3]. In the form (1), it has been considered in Bissantz and Birke [4] and Birke et al [5] with a deterministic design and in Hildebrandt et al [6] with a random design. These last works focus on kernel methods and establish their asymptotic normality.…”

Model Checks in Inverse Regression Models with Convolution‐Type Operators

“…, (x n,n , Y −n,n ) is a statistical inverse problem (see for example Bissantz et al (2007) or Mair and Ruymgaart (1996)). Estimation of the regression function f in model (1) and associated confidence bands have been discussed for Fourier based estimators in Bissantz and Birke (2009) and Birke et al (2009).…”