Rate-optimal nonparametric estimation in classical and Berkson errors-in-variables problems

We consider nonparametric estimation of a regression curve when the data are observed with Berkson errors or with a mixture of classical and Berkson errors. In this context, other existing nonparametric procedures can either estimate the regression curve consistently on a very small interval or require complicated inversion of an estimator of the Fourier transform of a nonparametric regression estimator. We introduce a new estimation procedure which is simpler to implement, and study its asymptotic properties. We derive convergence rates which are faster than those previously obtained in the literature, and we prove that these rates are optimal. We suggest a data-driven bandwidth selector and apply our method to some simulated examples.