Robust parameter estimation of density functions under fuzzy interval observations

Abstract

This study deals with the derivation of a probabilistic parametric model from interval data using the maximum likelihood principle. In contrast with classical techniques such as the EM algorithm, that define a precise likelihood function by computing the probability of observations viewed as a collection of non-elementary events, our approach presupposes that each imprecise observation underlies a precise one, and that the uncertainty that pervades its observation is epistemic, rather than representing noise. We define an interval-valued likelihood function and apply robust optimisation methods to find a safe plausible estimate of the statistical parameters. The approach is extended to fuzzy data by optimizing the average of lower likelikoods over a collection of data sets obtained from cuts of the fuzzy intervals, as a trade off between optimistic and pessimistic interpretations of fuzzy data. The principles of this method are compared with those of other existing approaches to handle incompleteness of observations, especially the EM technique.