Towards an interval-valued estimation of the density

Abstract

This paper presents a theoretical and practical novel approach for computing the probability density function underlying a set of observations. The estimator we propose is an extension of the conventional Parzen Rosenblatt method that leads to a very specific interval-valued estimation of the density. Within this approach, we make use of the convenient representation of a set of usual (summative) kernels by a maxitive kernel (i.e. a possibility distribution) to derive an exact computation with a very low complexity of an interval-valued estimation. The considered set of kernels is particularly convenient since it contains kernels having comparable shapes and bandwidth. We prove that the obtained imprecise probability density function contains a set of precise density functions estimated using the standard method with kernels belonging to the considered set.