Simultaneous inference for Berkson errors-in-variables regression under fixed design

In various applications of regression analysis, in addition to errors in the dependent observations also errors in the predictor variables play a substantial role and need to be incorporated in the statistical modeling process. In this paper we consider a nonparametric measurement error model of Berkson type with fixed design regressors and centered random errors, which is in contrast to much existing work in which the predictors are taken as random observations with random noise. Based on an estimator that takes the error in the predictor into account and on a suitable Gaussian approximation, we derive %uniform confidence statements for the function of interest. In particular, we provide finite sample bounds on the coverage error of uniform confidence bands, where we circumvent the use of extreme-value theory and rather rely on recent results on anti-concentration of Gaussian processes. In a simulation study we investigate the performance of the uniform confidence sets for finite samples

Nonparametric Berkson regression under normal measurement error and bounded design

Regression data often suffer from the so-called Berkson measurement error which contaminates the design variables. Conventional nonparametric approaches to this errors-in-variables problem usually require rather strong conditions on the support of the design density and that of the contaminated regression function, which seem unrealistic in many cases. In the current note, we introduce a novel nonparametric regression estimator, which is able to identify the regression function on the whole real line under normal Berkson error although the location of the design variables is restricted to some bounded interval. The asymptotic properties of this estimator are investigated and some numerical simulations are provided.Berkson error Deconvolution Errors-in-variables regression Inverse problems Orthogonal polynomials