Tsallis Entropy for Loss Models and Survival Models Involving Truncated and Censored Random Variables

Abstract

The aim of this paper consists in developing an entropy-based approach to risk assessment for actuarial models involving truncated and censored random variables by using the Tsallis entropy measure. The effect of some partial insurance models, such as inflation, truncation and censoring from above and truncation and censoring from below upon the entropy of losses is investigated in this framework. Analytic expressions for the per-payment and per-loss entropies are obtained, and the relationship between these entropies are studied. The Tsallis entropy of losses of the right-truncated loss random variable corresponding to the per-loss risk model with a deductible d and a policy limit u is computed for the exponential, Weibull, χ2 or Gamma distribution. In this context, the properties of the resulting entropies, such as the residual loss entropy and the past loss entropy, are studied as a result of using a deductible and a policy limit, respectively. Relationships between these entropy measures are derived, and the combined effect of a deductible and a policy limit is also analyzed. By investigating residual and past entropies for survival models, the entropies of losses corresponding to the proportional hazard and proportional reversed hazard models are derived. The Tsallis entropy approach for actuarial models involving truncated and censored random variables is new and more realistic, since it allows a greater degree of flexibility and improves the modeling accuracy.

Order Properties Concerning Tsallis Residual Entropy

“…Among the applications of other entropies (Rényi entropy, Varma entropy, Kaniadakis entropy, relative entropy, weighted entropy, etc. ), we can list the following: Markov chains (see [16][17][18]), model selection (see [19,20]), combinatorics (see [21,22]), finance (see [23][24][25]), Lie symmetries (see [26,27]), and machine learning (see [28,29]).…”

Information and Divergence Measures

“…Indeed, distances and entropies are related to inferential statistics, including both estimation and hypothesis testing problems [ 1 , 2 , 3 , 4 , 5 , 6 ], model selection criteria [ 7 , 8 , 9 ] and probabilistic and statistical modelling with applications in multivariate analysis, actuarial science, portfolio optimization, survival analysis, reliability theory, change-point problems, etc. [ 10 , 11 , 12 , 13 , 14 , 15 ]. Thus, the significance of entropy and divergence measures that emerges in these and many more scientific fields is a topic of great interest to scientists, researchers, medical experts, engineers, industrial managers, computer experts, data analysts, etc.…”