Nonparametric estimation of \({\mathbb {P}}(X from noisy data samples with non-standard error distributions

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