A note on an approximation and estimation of distribution function of difference of random variables

Abstract In this paper we deal with the problem of determining the distribution function of the difference of two independent random variables. Using a quantile-based representation we obtain an approximation of distribution function of difference of two independent random variables. Next, we obtain the error of this approximation. Finally, we use the approximation to present a non-parametric estimator for the distribution function of difference of two independent random variables. Moreover, we prove the strong consistency of this estimator and we carry out a numerical simulation to evaluate the bias and mean squared error of the estimator. Also we compare our estimator with the classical empirical distribution function.