Non-parametric regression estimation from data contaminated by a mixture of Berkson and classical errors

Abstract

Estimation of a regression function is a well-known problem in the context of errors in variables, where the explanatory variable is observed with random noise. This noise can be of two types, which are known as classical or Berkson, and it is common to assume that the error is purely of one of these two types. In practice, however, there are many situations where the explanatory variable is contaminated by a mixture of the two errors. In such instances, the Berkson component typically arises because the variable of interest is not directly available and can only be assessed through a proxy, whereas the inaccuracy that is related to the observation of the latter causes an error of classical type. We propose a non-parametric estimator of a regression function from data that are contaminated by a mixture of the two errors. We prove consistency of our estimator, derive rates of convergence and suggest a data-driven implementation. Finite sample performance is illustrated via simulated and real data examples. © 2007 Royal Statistical Society.