Ordering Properties of Extreme Claim Amounts From Heterogeneous Portfolios

Abstract

In the context of insurance, the smallest and largest claim amounts turn out to be crucial to insurance analysis since they provide useful information for determining annual premium. In this paper, we establish sufficient conditions for comparing extreme claim amounts arising from two sets of heterogeneous insurance portfolios according to various stochastic orders. It is firstly shown that the weak supermajorization order between the transformed vectors of occurrence probabilities implies the usual stochastic ordering between the largest claim amounts when the claim severities are weakly stochastic arrangement increasing. Secondly, sufficient conditions are established for the right-spread ordering and the convex transform ordering of the smallest claim amounts arising from heterogeneous dependent insurance portfolios with possibly different number of claims. In the setting of independent multiple-outlier claims, we study the effects of heterogeneity among sample sizes on the stochastic properties of the largest and smallest claim amounts in the sense of the hazard rate ordering and the likelihood ratio ordering. Numerical examples are provided to highlight these theoretical results as well. Not only can our results be applied in the area of actuarial science, but also they can be used in other research fields including reliability engineering and auction theory.

On fail‐safe systems under random shocks

“…However, their result is not correct since they neglected the value of the density ratio function at point t = 0. Fortunately, this could be corrected according to the proof method of Theorem ; see the work of Zhang et al…”

A Method for Constructing and Interpreting Some Weighted Premium Principles

“…Therefore, we refer them as record claims. Other recent studies focussing on stochastic properties of functions of extreme claims include Barmalzan et al (2016Barmalzan et al ( , 2017 and Zhang et al (2019). Let {X i } i≥1 be a sequence of independent claims having the same distribution as X and denote by F X and F X (x), respectively, the common distribution function and tail function.…”

Ordering results between the largest claims arising from two general heterogeneous portfolios

“…They proposed sufficient conditions to show various stochastic orderings between the largest claim amounts. Zhang et al [20] established conditions to compare the extreme claims from two collections of insurance portfolios. Barmalzan et al [1] developed stochastic comparisons results of extreme claim amounts having location-scale claim severities.…”