Imprecise Functional Estimation: The Cumulative Distribution Case

In this paper, we propose an adaptation of the Parzen Rosenblatt cumulative distribution function estimator that uses maxitive kernels. The result of this estimator, on every point of the domain of F , the cumulative distribution to be estimated, is interval valued instead of punctual valued. We prove the consistency of our approach with the classical Parzen Rosenblatt estimator, since, according to consistency conditions between the maxitive kernel involved in the imprecise estimator and the summative kernel involved in the precise estimator, our imprecise estimate contains the precise Parzen Rosenblatt estimate.