Beyond Traditional Interval Uncertainty in Describing Statistical Characteristics: Case of Interval Bounds on the Probability Density Function

Abstract

In the previous chapter, we considered the case when, in addition to knowing the interval bounds on the cumulative distribution function F(x) (= p-box), we also know that the function F(x) is smooth. In addition to knowing that F(x) is smooth – i.e., that its derivative F′(x) (= a probability density function) is bounded – we sometimes also know the bounds on F′(x). Such a situation is analyzed in this chapter.

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.

The results of this chapter have previously appeared in [354].