Nonparametric Estimation in a Multiplicative Censoring Model with Symmetric Noise

Abstract. We consider the model Yi “ XiUi, i “ 1, . . . , n, where the Xi, the Ui and thus the Yi are all independent and identically distributed. The Xi have density f and are the variables of interest, the Ui are multiplicative noise with uniform density on r1 ́ a, 1` as, for some 0 ă a ă 1, and the two sequences are independent. However, only the Yi are observed. We study nonparametric estimation of both the density f and the corresponding survival function. In each context, a projection estimator of an auxiliary function is built, from which estimator of the function of interest is deduced. Risk bounds in term of integrated squared error are provided, showing that the dimension parameter associated with the projection step has to perform a compromise. Thus, a model selection strategy is proposed in both cases of density and survival function estimation. The resulting estimators are proven to reach the best possible risk bounds. Simulation experiments illustrate the good performances of the estimators and a real data example is described.