On the cumulative residual interval entropy of doubly truncated random variables

1. Problem Statement and Motivation

The paper addresses the need for uncertainty measures in scenarios where data is doubly truncated (i.e., observations are only available within a specific interval $[\tau_1, \tau_2]$).

• Context: In fields like reliability engineering, survival analysis, actuarial science, and economics, systems are often observed only after they have survived a certain age ($\tau_1$) and before they fail or are censored at a later time ($\tau_2$).
• Limitation of Existing Measures:
  • Shannon Entropy ($S(X)$): Requires density estimation and can be negative.
  • Cumulative Residual Entropy (CRE, $E(X)$): Defined by Rao et al., this is non-negative but applies to the full support $[0, \infty)$.
  • Dynamic CRE ($E(X; t)$): Defined by Asadi and Zohrevand, this handles left-truncation (residual life after time $t$) but does not account for an upper bound (right truncation).
• Gap: There was no established measure for the uncertainty of a random variable $X$ conditioned on the event $\tau_1 \le X \le \tau_2$ that utilizes the cumulative survival function rather than the probability density function.

2. Methodology and Definitions

The authors introduce a new information-theoretic measure called the Cumulative Residual Interval Entropy (CRIE).

Key Definition

Let $X$ be a non-negative, absolutely continuous random variable with survival function $\bar{F}_X(x)$. The doubly truncated random variable is defined as $X_{\tau_1, \tau2} = X \mid \tau_1 \le X \le \tau_2$.
The CRIE, denoted $H(X; \tau_1, \tau_2)$, is defined as the cumulative residual entropy of this truncated variable:
$$H(X; \tau_1, \tau_2) = -\int_{\tau_1}^{\tau_2} \bar{F}_{X_{\tau_1, \tau_2}}(x) \ln \bar{F}_{X_{\tau_1, \tau_2}}(x) \, dx$$
where $\bar{F}_{X_{\tau_1, \tau_2}}(x) = \frac{\bar{F}_X(x) - \bar{F}_X(\tau_2)}{\bar{F}_X(\tau_1) - \bar{F}_X(\tau_2)}$.

Note: The authors explicitly correct a common misconception in literature (Remark 1) where previous studies used an incorrect survival function formulation for truncated variables. They define the measure based on the valid conditional survival function.

Auxiliary Concepts

To derive properties of CRIE, the paper utilizes:

• Generalized Failure Rate (GFR): $h_{X,1}(\tau_1, \tau_2) = \frac{f_X(\tau_1)}{\bar{F}_X(\tau_1) - \bar{F}_X(\tau_2)}$.
• Doubly Truncated Mean Residual Lifetime (DTMRL): $m_{X,1}(\tau_1, \tau_2) = E[X - \tau_1 \mid \tau_1 \le X \le \tau_2]$.
• Relevation Transform: A transformation used to relate entropy to the mean of transformed variables.

3. Key Contributions and Results

A. Fundamental Properties and Representations

The paper establishes several alternative representations for $H(X; \tau_1, \tau_2)$, linking it to established reliability measures:

1. Relation to DTMRL: $H(X; \tau_1, \tau_2) = E[m_{X,1}(X, \tau_2) \mid \tau_1 \le X \le \tau_2]$. This shows CRIE is the expected value of the mean residual lifetime of the truncated variable.
2. Covariance Representation: $H(X; \tau_1, \tau_2) = \text{Cov}(X, -\ln[\bar{F}_X(X) - \bar{F}_X(\tau_2)] \mid \tau_1 \le X \le \tau_2)$.
3. Relevation Transform: The CRIE is expressed as the difference between the mean of the relevation transform of the truncated variable and its DTMRL:
   $$H(X; \tau_1, \tau_2) = \int_{\tau_1}^{\tau_2} (\bar{F}_{X_{\tau_1, \tau_2}} \# \bar{F}_{X_{\tau_1, \tau_2}})(x) \, dx - m_{X,1}(\tau_1, \tau_2)$$
4. Transformation Invariance: The behavior of CRIE under increasing transformations $\phi(X)$ is analyzed, showing how the measure scales with linear transformations ($aX+b$).

B. Bounds and Inequalities

The authors derive rigorous upper and lower bounds for CRIE, which are crucial for practical estimation when analytical integration is difficult:

• Variance Bound: $H(X; \tau_1, \tau_2) \le \sqrt{\text{Var}(X \mid \tau_1 \le X \le \tau_2)}$.
• Mean Residual Lifetime Bounds:
  • Upper bound involving the logarithm of the ratio of the interval length to the DTMRL.
  • Lower bounds involving the interval entropy (Shannon entropy) and the DTMRL.
• Stochastic Ordering: If $X \le_{lr} Y$ (likelihood ratio order), specific inequalities relating $H(X)$ and $H(Y)$ are established, including a "Cumulative Residual Interval Kullback-Leibler (CRIKL)" divergence measure.

C. Monotonicity and New Distribution Classes

The paper introduces two new classes of distributions based on the monotonicity of CRIE with respect to the lower truncation point $\tau_1$:

1. ICRIE (Increasing CRIE): $H(X; \tau_1, \tau_2)$ increases as $\tau_1$ increases.
2. DCRIE (Decreasing CRIE): $H(X; \tau_1, \tau_2)$ decreases as $\tau_1$ increases.

Key Theorems on Monotonicity:

• Characterization: $X$ is ICRIE (DCRIE) if and only if $H(X; \tau_1, \tau_2) \ge (\le) m_{X,1}(\tau_1, \tau_2)$.
• Closure Properties:
  • The ICRIE/DCRIE classes are closed under increasing linear transformations.
  • The DCRIE class is closed under increasing convex transformations.
• Relation to Failure Rates: If the Generalized Failure Rate $h_{X,1}$ is decreasing (increasing), then $X$ belongs to the DCRIE (ICRIE) class.

4. Significance and Applications

• Theoretical Advancement: The paper generalizes the Cumulative Residual Entropy (CRE) and Dynamic CRE to the doubly truncated case, providing a mathematically rigorous framework for interval-based uncertainty.
• Practical Utility:
  • Reliability: Essential for analyzing systems with inspection intervals or warranty periods where failure times are only known to lie within a window.
  • Actuarial Science: Directly applicable to insurance models involving franchise deductibles and policy limits (e.g., losses between $d$ and $u$).
  • Goodness-of-Fit: The proposed CRIKL divergence offers a new statistical tool for testing hypotheses about distribution functions in truncated data scenarios.
• Computational Feasibility: By providing bounds based on variance and mean residual life, the paper offers practical ways to estimate uncertainty without needing complex density estimations.

Conclusion

Chadjiconstantinidis and Bozikas successfully define and characterize the Cumulative Residual Interval Entropy (CRIE). They demonstrate that this measure is a robust, non-negative uncertainty metric that generalizes existing entropy measures. Through the establishment of bounds, stochastic ordering properties, and new distribution classes (ICRIE/DCRIE), the paper provides a comprehensive toolkit for researchers and practitioners dealing with doubly truncated data in reliability and risk analysis.