As a natural extension of the fuzzy variable, bifuzzy variable is defined as mapping from credibility space to collection variables, which an appropriate tool model two-fold phenomena. In order enrich its theoretical foundation, this paper explores some important measures for regular most commonly used type variables. Firstly, we introduce variables’ mean chance measure and properties, includin...

The earth mover's distance (EMD), also called the first Wasserstein distance, can be naturally extended to compare arbitrarily many probability distributions, rather than only two, on set $[n]=\{1,\dots,n\}$. We present details for this generalization, along with a highly efficient algorithm inspired by combinatorics; it turns out that in special case of three EMD is half sum pairwise EMD's. Ex...

Abstract Conditional value-at-risk (CVaR) and conditional expected shortfall (CES) are widely adopted risk measures which help monitor potential tail while adapting to evolving market information. In this paper, we propose an approach constructing simultaneous confidence bands (SCBs) for as measured by CVaR CES, with the uniformly valid a set of levels. We consider one-sided (downside or upside...