Allocations of policy limits and ordering relations for aggregate remaining claims

Suppose that a policyholder faces n risks X 1 , . . . , X n which are insured under the policy limit with the total limit of l. Usually, the policyholder is asked to protect each X i with an arbitrary limit of l i such that n i=1 l i = l. If the risks are independent and identically distributed with log-concave cumulative distribution function, using the notions of majorization and stochastic orderings, we prove that the equal limits provide the maximum of the expected utility of the wealth of policyholder. If the risks with log-concave distribution functions are independent and ordered in the sense of the reversed hazard rate order, we show that the equal limits is the most favourable allocation among the worst allocations. We also prove that if the joint probability density function is arrangement increasing, then the best arranged allocation maximizes the utility expectation of policyholder's wealth. We apply the main results to the case when the risks are distributed according to a log-normal distribution.

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“…, 0), respectively (Corollaries 2.2 and 2.5). These are extensions of the results of Theorem 2.4 and 2.8 in Manesh and Khaledi (2015).…”

Section: Definition 15 a Real Valued Function φ Defined On Setsupporting

confidence: 70%

Some New Results on Policy Limit Allocations

Manesh

Izadi

Khaledi

2021

JIRSS

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“…Xu and Hu (2011) extended this ordering result to independent but not necessarily identically distributed random variables with some regularity conditions. For more on ordering properties of standard linear combinations of independent random variables, we refer the reader to Bock et al (1987), Kijima and Ohnishi (1996), Ma (2000), Hu and Lin (2001), Khaledi and Kochar (2001), Korwar (2002), Khaledi and Kochar (2004), Nadarajah and Kotz (2005), Manesh and Khaledi (2008), Amiri et al (2011), Kochar and Xu (2010), (2011), Zhao (2011), Zhao et al (2011), Manesh and Khaledi (2015) and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…. , d n ) denotes the corresponding parameters such as deductibles, coverage limits, allocated capitals, and so on, and φ (x, d) is a bivariate function measuring the consequence of matching d to x; see, for example, Cheung and Yang (2004), Cheung (2006), Li and You (2015), and Manesh and Khaledi (2015).…”

Section: Introductionmentioning

confidence: 99%

Increasing convex order on generalized aggregation of SAI random variables with applications

Pan

2017

J. Appl. Probab.

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In this paper we study general aggregation of stochastic arrangement increasing random variables, including both the generalized linear combination and the standard aggregation as special cases. In terms of monotonicity, supermodularity, and convexity of the kernel function, we develop several sufficient conditions for the increasing convex order on the generalized aggregations. Some applications in reliability and risks are also presented.

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“…In the context of insurance, X : = ( X 1 ,…, X n ) may denote a set of potential risks/losses, a : = ( a 1 ,…, a n ) represents the corresponding parameters such as allocated capitals, deductibles, policy limits, and so on, and

ϕ

is a bivariate function with many forms arising from practical situations, such as

ϕ (x, a) = {(x - a)}_{+}

ϕ (x, a) = \min {x, a}

and

ϕ (x, a) = ψ (x - a)

for some convex function

ψ

, respectively. For the study on the ordering properties of

\sum_{i = 1}^{n} ϕ (X_{i}, a_{i})

with respect to the usual stochastic ordering and the increasing convex ordering, interested readers are referred to References 11‐14.…”

Section: Introductionmentioning

confidence: 99%

Ordering properties of generalized aggregation with applications

Ding

Wang

Zhang

2020

Appl Stoch Models Bus & Ind

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The generalized aggregation ∑i=1nWiϕ(Xi,ai) arises in many research fields including applied probability, actuarial science, and reliability theory, where ϕ is a bivariate kernel function and a is a parameter vector. One of its remarkable features is that both X and W are dependent in many practical situations. Therefore, studying the stochastic properties of generalized aggregations under various dependence structures is an interesting and meaningful problem. In this paper, by using left tail weakly stochastic arrangement increasing, right tail weakly stochastic arrangement increasing, and comonotonicity to characterize the dependent structures among X or W, we establish the increasing convex ordering and the expectation ordering of generalized aggregations to investigate the effects of the arrangement and heterogeneity among ai's. Numerical examples and three practical applications are presented to illustrate our results as well.

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Allocations of policy limits and ordering relations for aggregate remaining claims

Cited by 10 publications

References 12 publications

Some New Results on Policy Limit Allocations

Some New Results on Policy Limit Allocations

Increasing convex order on generalized aggregation of SAI random variables with applications

Ordering properties of generalized aggregation with applications

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