Doubly truncated (interval) cumulative residual and past entropy

Two measures of uncertainty, the cumulative residual entropy (CRE) and doubly truncated Shannon entropy have been introduced respectively by Rao et al. (2004) and Sunoj et al. (2009). In this paper we proposed the doubly truncated (interval) cumulative residual entropy (ICRE), which is an extension of CRE and the dynamic CRE defined by Asadi and Zohrevand (2007). We study some properties and characterization of this measure, including its connections with doubly truncated Shannon entropy and mean residual life. Also, the dual measure, doubly truncated (interval) cumulative past entropy (ICPE) is defined and some of its properties are studied. Furthermore, their monotonicity and associated aging classes of distributions are discussed and the upper (lower) bound for them are obtained. Finally for exponential distribution all entropies are presented.