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On the cumulative residual interval entropy of doubly truncated random variables

Stathis Chadjiconstantinidis, Apostolos Bozikas

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.

On the cumulative residual interval entropy of doubly truncated random variables

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.
Paper Structure (5 sections, 37 theorems, 206 equations, 2 figures, 1 table)

This paper contains 5 sections, 37 theorems, 206 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Some results on the generalized failure rate and doubly truncated mean residual lifetime
  3. Representations for $\boldsymbol{H(X;\tau_1,\tau_2)}$
  4. Bounds for $\boldsymbol{H(X;\tau_1,\tau_2)}$
  5. Monotonicity results and further bounds for $\boldsymbol{H(X;\tau_1,\tau_2)}$

Key Result

Lemma 1

For any $(\tau_1, \tau_2) \in D_X$ it holds (i) (ii)

Figures (2)

  • Figure 1: Plots of the mean residual lifetime function $m_{X,1}(\tau_1,\tau_2)$ (left panel) and the entropy measure $H(X; \tau_1, \tau_2)$ (right panel) versus $\tau_1$ with fixed $\tau_2 = 0.9$ for the beta distribution with $\overline{F}(x) = 1 - x^c$ for $0 < x < 1$, where $c = 1, 2, 5$.
  • Figure 2: Plots of the mean residual lifetime function $m_{X,1}(\tau_1,\tau_2)$ (left panel) and the entropy measure $H(X; \tau_1, \tau_2)$ (right panel) versus $\tau_1$ with fixed $\tau_2 = 10$ for the exponential distribution with $\overline{F}(x) = \exp(-\lambda x)$ for $x \geq 0$, where $\lambda = 0.2, 0.5, 1$.

Theorems & Definitions (83)

  • Definition 1
  • Remark 1
  • Remark 2
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 73 more