Nonparametric estimation of P(X<Y) 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 θ : = P ( X < Y ) based on independent random samples from the distributions of X ′ , Y ′ , ζ and η , where X ′ = X + ζ , Y ′ = Y + η and X , Y , ζ , η are mutually independent random variables. In this context, ζ , η are referred to as measurement errors. We apply the ridge-parameter regularization method to derive a nonparametric estimator for θ depending on two parameters. Our estimator is shown to be consistent with respect to mean squared error if the characteristic functions of ζ , η only vanish on Lebesgue measure zero sets. Under some further assumptions on the densities of X , Y , ζ and η , 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.