An Estimation of P(X<Y<Z)Using RepeatedObservations with Unknown Noise Distribution

We study the problem of estimating the probability theta:=P(X<Y<Z)whenthe variablesX,Zfollow the Gaussian distributions with known parameters and thevariableYis distributed with an unknown distribution. Our available data are somereplicated contaminated measurements onYcontaining some underlying noises, inwhich the common distribution of the noises are completely unknown, but it is sym-metric around zero. Based on the available data as well as on the full knowledge aboutthe distributions ofX,Z, we propose a nonparametric estimator of theta in presence ofone regularization parameter. Under some regularity assumptions on the distributionofYand the noise distribution, we derive some rates of convergence of the proposedestimator with respect to the mean squared error. Several numerical experiments arealso conducted in order to illustrate the convergence of our estimator.