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)\documentclass[12pt]{minimal}
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\begin{document}$$\theta := {\mathbb {P}}(X<Y)$$\end{document} based on independent random samples from the distributions of X′\documentclass[12pt]{minimal}
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\begin{document}$$X'$$\end{document}, Y′\documentclass[12pt]{minimal}
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\begin{document}$$Y'$$\end{document}, ζ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta $$\end{document} and η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document}, where X′=X+ζ\documentclass[12pt]{minimal}
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\begin{document}$$X' = X + \zeta $$\end{document}, Y′=Y+η\documentclass[12pt]{minimal}
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\begin{document}$$Y' = Y + \eta $$\end{document} and X, Y, ζ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta $$\end{document}, η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document} are mutually independent random variables. In this context, ζ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta $$\end{document}, η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document} are referred to as measurement errors. We apply the ridge-parameter regularization method to derive a nonparametric estimator for θ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document} depending on two parameters. Our estimator is shown to be consistent with respect to mean squared error if the characteristic functions of ζ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta $$\end{document}, η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document} only vanish on Lebesgue measure zero sets. Under some further assumptions on the densities of X, Y, ζ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta $$\end{document} and η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document}, 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.
