Beyond Interval Uncertainty in Describing Statistical Characteristics: Case of Smooth Distributions and Info-Gap Decision Theory

Abstract

In the traditional statistical approach, we assume that we know the exact cumulative distribution function (CDF) F(x). In practice, we often only know the envelopes [\(\underline{F}(x), \overline{F}(x)\)] bounding this CDF, i.e., we know the intervalvalued “p-box” which contains F(x). P-boxes have been successfully applied to many practical applications. In the p-box approach, we assume that the actual CDF can be any CDF \(F(x) \epsilon [\underline{F}(x), \overline{F}(x)\)]. In many practical situations, however, we know that the actual distribution is smooth. In such situations, we may wish our model to further restrict the set of CDFs by requiring them to share smoothness (and similar) properties with the bounding envelopes \(\underline{F}(x)\) and \(\overline{F}(x)\). In previous work, ideas from Info-Gap Decision Theory were used to propose heuristic methods for selecting such distributions. In this chapter, we provide justifications for this heuristic approach.The main results of this chapter first appeared in [38].