We study the problem of estimating the probability \(\theta := \mathbb {P}(X<Y<Z)\) when the variables X, Z follow the Gaussian distributions with known parameters and the variable Y is distributed with an unknown distribution. Our available data are some replicated contaminated measurements on Y containing some underlying noises, in which the common distribution of the noises are completely unknown, but it is symmetric around zero. Based on the available data as well as on the full knowledge about the distributions of X, Z, we propose a nonparametric estimator of \(\theta \) in presence of one regularization parameter. Under some regularity assumptions on the distribution of Y and the noise distribution, we derive some rates of convergence of the proposed estimator with respect to the mean squared error. Several numerical experiments are also conducted in order to illustrate the convergence of our estimator.
