Adaptive estimation of the hazard rate with multiplicative censoring

We propose an adaptive estimation procedure of the hazard rate of a random variable X in the multiplicative censoring model, Y=XU, with U∼U([0,1]) independent of X. The variable X is not directly observed: an estimator is built from a sample {Y1,...,Yn} of copies of Y. It is obtained by minimisation of a contrast function over a class of general nested function spaces which can be generated e.g. by splines functions. The dimension of the space is selected by a penalised contrast criterion. The final estimator is proved to achieve the best bias-variance compromise and to reach the same convergence rate as the oracle estimator under conditions on the maximal dimension. The good behavior of the resulting estimator is illustrated over a simulation study.