Confidence bands for inverse regression models

Abstract

We construct uniform confidence bands for the regression function in inverse, homoscedastic regression models with convolution-type operators. Here, the convolution is between two non-periodic functions on the whole real line rather than between two periodic functions on a compact interval, since the former situation arguably arises more often in applications. First, following Bickel and Rosenblatt (1973 Ann. Stat. 1 1071-95) we construct asymptotic confidence bands which are based on strong approximations and on a limit theorem for the supremum of a stationary Gaussian process. Further, we propose bootstrap confidence bands based on the residual bootstrap and prove consistency of the bootstrap procedure. A simulation study shows that the bootstrap confidence bands perform reasonably well for moderate sample sizes. Finally, we apply our method to data from a gel electrophoresis experiment with genetically engineered neuronal receptor subunits incubated with rat brain extract.

Exponential Inequalities in Functional Nonparametrics Regression for Mixing Process

“…The observations Z k are either fixed points (ex. M. Birke, N. Bissantz et H. Holzman [2]) or Z k are random variables (ex. L. Desol [4], A. Tadj [5]).…”

“…Since then, their methods were much developed in the context of estimation of density function and regression function. M. Birke, N. Bissantz and H. Holzman [2] Have constructed uniform confidence intervals for an inverse regression function in a similar way to Bickel and Rosenblatt, based on approximations and the limit theorem of a stationary Gaussian process. Recently, L. Desole [3] studies a generalization of the Nadaraya-Watson type estimators in the case of the α-mixing functional variables…”

Comments on: An updated review of Goodness-of-Fit tests for regression models

“…The function m is not directly observable and the problem of estimating m has found considerable interest in the recent statistical literature (see for example Mair and Ruymgaart 1996;Kaipio and Somersalo 2010;Cavalier 2008;Bertero et al 2009or Birke et al 2010 among others). In inverse regression the simplification of the statistical analysis obtained by parametric models is even more substantial than in the direct case and consequently goodness-of-fit testing is particularly relevant in this context.…”

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).…”