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

Let X, Y, W, $$\delta $$ δ and $$\varepsilon $$ ε be continuous univariate random variables defined on a probability space such that $$Y = X+\varepsilon $$ Y = X + ε and $$W = X + \delta $$ W = X + δ . Herein X, $$\delta $$ δ and $$\varepsilon $$ ε are assumed to be mutually independent. The variables $$\varepsilon $$ ε and $$\delta $$ δ are called classical and Berkson errors, respectively. Their distributions are known exactly. Suppose we only observe a random sample $$Y_1, \ldots , Y_n$$ Y 1 , … , Y n from the distribution of Y. This paper is devoted to a nonparametric estimation of the unknown cumulative distribution function $$F_W$$ F W of W based on the observations as well as on the distributions of $$\varepsilon $$ ε , $$\delta $$ δ . An estimator for $$F_W$$ F W 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, $$\delta $$ δ and $$\varepsilon $$ 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.