On the cumulative residual interval entropy of doubly truncated random variables

TL;DR

This paper introduces the cumulative residual interval entropy (CRIE), a new measure of uncertainty for doubly truncated continuous random variables, with theoretical properties and bounds explored.

Contribution

It generalizes existing entropy measures and provides new properties, representations, and bounds for CRIE in the context of doubly truncated data.

Findings

CRIE generalizes previous entropy measures.

Properties of the generalized hazard rate are established.

Bounds and monotonicity results for CRIE are derived.

Abstract

This paper introduces and studies a new uncertainty measure, the cumulative residual interval entropy (CRIE). Defined as the cumulative residual entropy of a doubly truncated (interval) continuous random variable, this measure has several applications when data fall between two points. The CRIE generalizes the cumulative residual entropy proposed by Rao et al. [31] and the dynamic cumulative residual entropy proposed by Asadi and Zohrevand [1]. We establish some properties of the generalized hazard rate and the doubly truncated mean residual lifetime, which are useful for obtaining results for the CRIE. Furthermore, we provide several representations of the CRIE based on reliability measures, covariance, the relevation transform, and increasing transformations. Finally, upper and lower bounds, as well as monotonicity results for the CRIE, are provided.