Nonparametric estimation of $${\mathbb {P}}(X

Abstract

Let X, Y be continuous random variables with unknown distributions. The aim of this paper is to study the problem of estimating the probability \(\theta := {\mathbb {P}}(X<Y)\) based on independent random samples from the distributions of \(X'\), \(Y'\), \(\zeta \) and \(\eta \), where \(X' = X + \zeta \), \(Y' = Y + \eta \) and X, Y, \(\zeta \), \(\eta \) are mutually independent random variables. In this context, \(\zeta \), \(\eta \) are referred to as measurement errors. We apply the ridge-parameter regularization method to derive a nonparametric estimator for \(\theta \) depending on two parameters. Our estimator is shown to be consistent with respect to mean squared error if the characteristic functions of \(\zeta \), \(\eta \) only vanish on Lebesgue measure zero sets. Under some further assumptions on the densities of X, Y, \(\zeta \) and \(\eta \), we obtain some upper and lower bounds on the convergence rate of the estimator. A numerical example is also given to illustrate the efficiency of our method.