Nonparametric estimation of cumulative distribution function from noisy data in the presence of Berkson and classical errors

Let X, Y, W, δ and ε be continuous univariate random variables defined on a probability space such that Y= X+ ε and W= X+ δ. Herein X, δ and ε are assumed to be mutually independent. The variables ε and δ are called classical and Berkson errors, respectively. Their distributions are known exactly. Suppose we only observe a random sample Y, … , Y from the distribution of Y. This paper is devoted to a nonparametric estimation of the unknown cumulative distribution function F of W based on the observations as well as on the distributions of ε, δ. An estimator for F depending on a smoothing parameter is suggested. It is shown to be consistent with respect to the mean squared error. Under certain regularity assumptions on the densities of X, δ and ε, we establish some upper and lower bounds on the convergence rate of the proposed estimator. Finally, we perform some numerical examples to illustrate our theoretical results.