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

设X , Y , W , \(\delta \)和\(\varepsilon \)是在概率空间上定义的连续单变量随机变量，使得\(Y = X+\varepsilon \)和\(W = X + \delta \)。这里假设X , \(\delta \)和\(\varepsilon \)是相互独立的。变量\(\varepsilon \)和\(\delta \)分别称为经典误差和伯克森误差。它们的分布是准确已知的。假设我们只从Y的分布中观察到一个随机样本\(Y_1, \ldots , Y_n\). 本文致力于基于观测值以及\(\varepsilon \) , \(\delta \)的分布对W的未知累积分布函数\(F_W\)进行非参数估计。建议使用取决于平滑参数的\(F_W\)估计器。它被证明在均方误差方面是一致的。在X、\(\delta \)和\(\varepsilon \)的密度的某些规律性假设下，我们建立了所提出的估计器收敛速度的一些上限和下限。最后，我们执行一些数值例子来说明我们的理论结果。