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ORDERING PROPERTIES OF EXTREME CLAIM AMOUNTS FROM HETEROGENEOUS PORTFOLIOS

Published online by Cambridge University Press:  29 April 2019

Yiying Zhang ,
Xiong Cai  and
Peng Zhao Open the ORCID record for Peng Zhao [Opens in a new window]
Yiying Zhang
Affiliation:
School of Statistics and Data Science, LPMC and KLMDASRNankai UniversityTianjin 300071, P.R. China E-Mail: [email protected]
Xiong Cai
Affiliation:
College of Applied SciencesBeijing University of TechnologyBeijing 100124, P.R. China E-Mail: [email protected]
Peng Zhao*
Affiliation:
School of Mathematics and StatisticsJiangsu Normal UniversityXuzhou 221116, P.R. China E-Mail: [email protected]
*
E-Mail: [email protected]
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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.

Keywords

Largest claim amountssmallest claim amountsstochastic ordersoccurrence probabilitiesmajorization.
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 49 , Issue 2 , May 2019 , pp. 525 - 554
Copyright
Copyright © Astin Bulletin 2019 

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