Two Sufficient Conditions for Convex Ordering on Risk Aggregation

Abstract and Applied Analysis, Feb 2018

We define new stochastic orders in higher dimensions called weak correlation orders. It is shown that weak correlation orders imply stop-loss order of sums of multivariate dependent risks with the same marginals. Moreover, some properties and relations of stochastic orders are discussed.

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Two Sufficient Conditions for Convex Ordering on Risk Aggregation

Two Sufficient Conditions for Convex Ordering on Risk Aggregation Dan Zhu and Chuancun Yin School of Statistics, Qufu Normal University, Shandong 273165, China Correspondence should be addressed to Chuancun Yin; nc.ude.unfq.liam@niycc Received 22 October 2017; Accepted 28 December 2017; Published 1 February 2018 Academic Editor: Lucas Jodar Copyright © 2018 Dan Zhu and Chuancun Yin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We define new stochastic orders in higher dimensions called weak correlation orders. It is shown that weak correlation orders imply stop-loss order of sums of multivariate dependent risks with the same marginals. Moreover, some properties and relations of stochastic orders are discussed. 1. Introduction Correlation order as an important stochastic order relation was first introduced by Joe [1]; Dhaene and Goovaerts [2] studied the bivariate case with the same marginals. After that, the bivariate case has been generalized by Lu and Zhang [3]. Recall that given two random vectors and with the same marginals, is said to be less correlated than , written as , if for every pair of disjoint subsets and of ,where and are nondecreasing functions for which the covariances exist. The main result of Lu and Zhang [3] showed that the correlation order implied stop-loss order for portfolios of multivariate dependent risks; that is, implies . Stop-loss order as a special case of convex order is the most frequently used order relation for the comparison of risks, written as , for any two random variables and , if and only if the inequality holds for all real , where denotes the positive part of the real . In addition, is said to precede in the convex order sense; define , if and only if and More details and other characterizations about stop-loss order can be found in Denuit et al. [4], Dhaene et al. [5], Landsman and Tsanakas [6], and Shaked and Shanthikumar [7]. Rüschendorf [8] introduced a new dependence order among risks called the weakly conditional increasing in sequence order; by definition given two random vectors and with the same marginals, is said to be smaller than in the weakly conditional increasing in sequence order, written as , if for all , all and is monotonically nondecreasing,where and . It is showed that more positive dependence with respect to the wcs ordering implied more risk with respect to the supermodular ordering (see Rüschendorf [8] for the definitions of supermodular function and supermodular ordering) for -dimensional random vectors; that is, implies . Note that, in (1), we let and , (2) is an immediate consequence of (1), and by Example 5 of this paper, the weakly conditional increasing in sequence order is weaker than correlation order, while the stop-loss order still holds by Müller [9]. Enlightened by this, we are committed to find more general conditions for the multivariate case which can also imply stop-loss order. In this short note, we give the concepts of weak correlation orders in higher dimensions and show that the weak correlation orders imply stop-loss order of multivariate dependent risks with the same marginals. The remainder of the paper is organized as follows. In Section 2, we introduce some concepts of stochastic orders including the new definitions and discuss the properties and stochastic order relations. The main results of this paper are presented and proved in Section 3. 2. Preliminaries Given Fréchet space of all -dimensional random vectors , we have as marginal distributions, where , and the joint distribution function is For all , we have the following inequality: where and are called Fréchet upper bound and Fréchet lower bound of , respectively. Remark that is reachable in and when , is indeed a distribution function. However, when , is no longer always a distribution function (see Denuit et al. [4]). A necessary and sufficient condition for to be a distribution function in can be found in Dhaene and Denuit [10]. Throughout the short note, it is assumed that all random variables are real random variables on this space. Comparing random variables is the essence of the actuarial profession; in order to acquire more general results, we give the notion of weak correlation orders as follows. Definition 1. Let random vectors and be elements of , and we say that is smaller than in type I weak correlation order, written as , if for all , , any of the following equivalent conditions holds:where is an indicator function. Remark 2. The equivalence between (5) can be obtained as follows. We haveand by the same way, we obtain Hence, (5) are equivalent. Moreover, (8) and (9) are also equivalent. Definition 3. Random vectors and are elements of , and we say that is smaller than in type II weak correlation order, written as , (...truncated)


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Dan Zhu, Chuancun Yin. Two Sufficient Conditions for Convex Ordering on Risk Aggregation, Abstract and Applied Analysis, 2018, 2018, DOI: 10.1155/2018/2937895