Computing Statistical Characteristics When We Know Probabilities with Interval or Fuzzy Uncertainty: Computational Complexity

In traditional statistics, we usually assume that we know the exact probability distributions. In practice, we often only know the probabilities with interval uncertainty. The main emphasis on taking this uncertainty into account has been on situations in which we know a cumulative distribution function (cdf) with interval uncertainty. However, in some cases, we know the probability density function (pdf) with interval uncertainty. We show that in this situations, the exact range of some statistical characteristics can be efficiently computed. Surprisingly, for some other characteristics, similar statistical problems which are efficiently solvable for interval-valued cdf become computationally difficult (NP-hard) for interval-valued pdf.