scholarly journals Two Sufficient Conditions for Convex Ordering on Risk Aggregation

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Dan Zhu ◽  
Chuancun Yin
Keyword(s):  
Sufficient Conditions ◽  
Higher Dimensions ◽  
Stochastic Orders ◽  
Weak Correlation ◽  
Convex Ordering ◽  
Risk Aggregation ◽  
Dependent Risks ◽  
Stop Loss

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.

ON RELATIVE AGING COMPARISONS OF COHERENT SYSTEMS WITH IDENTICALLY DISTRIBUTED COMPONENTS

2020 ◽  
pp. 1-15 ◽  
Author(s):  
Nil Kamal Hazra ◽  
Neeraj Misra
Keyword(s):  
Hazard Rate ◽  
Sufficient Conditions ◽  
Stochastic Orders ◽  
Coherent System ◽  
Coherent Systems ◽  
Cumulative Hazard ◽  
Numerical Examples ◽  
Distributed Components ◽  
Rate Functions ◽  
Important Notion

The relative aging is an important notion which is useful to measure how a system ages relative to another one. Among the existing stochastic orders, there are two important orders describing the relative aging of two systems, namely, aging faster orders in the cumulative hazard and the cumulative reversed hazard rate functions. In this paper, we give some sufficient conditions under which one coherent system ages faster than another one with respect to the aforementioned stochastic orders. Further, we show that the proposed sufficient conditions are satisfied for k-out-of-n systems. Moreover, some numerical examples are given to illustrate the applications of proposed results.

Stochastic Orders in Terms of Laplace Transforms

1995 ◽  
Vol 45 (3-4) ◽  
pp. 195-202 ◽  
Author(s):  
Asok K. Nanda
Keyword(s):  
Laplace Transform ◽  
Sufficient Conditions ◽  
Laplace Transforms ◽  
Necessary And Sufficient Conditions ◽  
Stochastic Orders ◽  
Partial Orderings ◽  
Life Distributions ◽  
Necessary And Sufficient

Recently s-FR and s-ST orderings have been defined in the literature. They are more general in the sense that most of the earlier known partial orderings reduce as particular cases of these orderings. Moreover, these orderings have helped in defining new and useful ageing criterion. In this paper, using Laplace transform, we characterize, by means of necessary and sufficient conditions. the property that two life distributions are ordered in the s-FR and s-ST sense. The characterization of LR, FR, MR, VR, STand HAMR orderings follow as particular cases.

Asymptotic stability of heteroclinic cycles in systems with symmetry

1995 ◽  
Vol 15 (1) ◽  
pp. 121-147 ◽  
Author(s):  
Martin Krupa ◽  
Ian Melbourne
Keyword(s):  
Asymptotic Stability ◽  
Ad Hoc ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Higher Dimensions ◽  
Systematic Investigation ◽  
Heteroclinic Cycles ◽  
General Sufficient Condition ◽  
Higher Dimensional ◽  
The Stability

AbstractSystems possessing symmetries often admit heteroclinic cycles that persist under perturbations that respect the symmetry. The asymptotic stability of such cycles has previously been studied on an ad hoc basis by many authors. Sufficient conditions, but usually not necessary conditions, for the stability of these cycles have been obtained via a variety of different techniques.We begin a systematic investigation into the asymptotic stability of such cycles. A general sufficient condition for asymptotic stability is obtained, together with algebraic criteria for deciding when this condition is also necessary. These criteria are always satisfied in ℝ3 and often satisfied in higher dimensions. We end by applying our results to several higher-dimensional examples that occur in mode interactions with O(2) symmetry.

ORDERING PROPERTIES OF EXTREME CLAIM AMOUNTS FROM HETEROGENEOUS PORTFOLIOS

2019 ◽  
Vol 49 (2) ◽  
pp. 525-554 ◽  
Author(s):  
Yiying Zhang ◽  
Xiong Cai ◽  
Peng Zhao
Keyword(s):  
Sufficient Conditions ◽  
Stochastic Ordering ◽  
Stochastic Orders ◽  
Reliability Engineering ◽  
Research Fields ◽  
Likelihood Ratio Ordering ◽  
Hazard Rate Ordering ◽  
Arrangement Increasing ◽  
The Right ◽  
Stochastic Properties

AbstractIn 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.

SUFFICIENT CONDITIONS FOR INTERPOLATION AND SAMPLING HYPERSURFACES IN THE BERGMAN BALL

2007 ◽  
Vol 18 (05) ◽  
pp. 559-584 ◽  
Author(s):  
TAMÁS FORGÁCS ◽  
DROR VAROLIN
Keyword(s):  
Recent Work ◽  
Fock Space ◽  
Sufficient Conditions ◽  
Higher Dimensions ◽  
Sampling Theorem ◽  
Interpolation Theorem ◽  
Smooth Hypersurface ◽  
Present Setting ◽  
Geometric Density ◽  
Natural Method

We give sufficient conditions for a closed smooth hypersurface W in the n-dimensional Bergman ball to be interpolating or sampling. As in the recent work [5] of Ortega-Cerdà, Schuster and the second author on the Bargmann–Fock space, our sufficient conditions are expressed in terms of a geometric density of the hypersurface that, though less natural, is shown to be equivalent to Bergman ball analogs of the Beurling-type densities used in [5]. In the interpolation theorem we interpolate L2 data from W to the ball using the method of Ohsawa–Takegoshi, extended to the present setting, rather than the Cousin I approach used in [5]. In the sampling theorem, our proof is completely different from [5]. We adapt the more natural method of Berndtsson and Ortega-Cerdà [1] to higher dimensions. This adaptation motivated the notion of density that we introduced. The approaches of [5] and the present paper both work in either the case of the Bergman ball or of the Bargmann–Fock space.

scholarly journals Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures

2007 ◽  
Vol 37 (1) ◽  
pp. 93-112 ◽  
Author(s):  
Jun Cai ◽  
Ken Seng Tan
Keyword(s):  
Value At Risk ◽  
Negative Binomial ◽  
Risk Model ◽  
Sufficient Conditions ◽  
Individual Risk ◽  
Risk Models ◽  
Optimization Criteria ◽  
Collective Risk Model ◽  
Collective Risk ◽  
Stop Loss

We propose practical solutions for the determination of optimal retentions in a stop-loss reinsurance. We develop two new optimization criteria for deriving the optimal retentions by, respectively, minimizing the value-at-risk (VaR) and the conditional tail expectation (CTE) of the total risks of an insurer. We establish necessary and sufficient conditions for the existence of the optimal retentions for two risk models: individual risk model and collective risk model. The resulting optimal solution of our optimization criterion has several important characteristics: (i) the optimal retention has a very simple analytic form; (ii) the optimal retention depends only on the assumed loss distribution and the reinsurer’s safety loading factor; (iii) the CTE criterion is more applicable than the VaR criterion in the sense that the optimal condition for the former is less restrictive than the latter; (iv) if optimal solutions exist, then both VaR- and CTE-based optimization criteria yield the same optimal retentions. In terms of applications, we extend the results to the individual risk models with dependent risks and use multivariate phase type distribution, multivariate Pareto distribution and multivariate Bernoulli distribution to illustrate the effect of dependence on optimal retentions. We also use the compound Poisson distribution and the compound negative binomial distribution to illustrate the optimal retentions in a collective risk model.

scholarly journals Increasing Mean Inactivity Time Ordering: A Quantile Approach

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
M. Kayid ◽  
S. Izadkhah ◽  
A. Alfifi
Keyword(s):  
Stochastic Order ◽  
Sufficient Conditions ◽  
Parallel Systems ◽  
The Other ◽  
Stochastic Orders ◽  
General Family ◽  
Order Relations ◽  
Inactivity Time ◽  
Time Order ◽  
Mean Inactivity Time

We study further the quantile mean inactivity time order. Relations between the proposed stochastic order and the other transform stochastic orders are obtained. Besides, sufficient conditions for the stochastic order are provided. Then, preservation of the order under monotone transformations, series, and parallel systems and mixtures of a general family of semiparametric distributions is studied. Examples are also given to illustrate the results.

scholarly journals Sufficient Conditions for Some Stochastic Orders of Discrete Random Variables with Applications in Reliability

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 147
Author(s):  
Félix Belzunce ◽  
Carolina Martínez-Riquelme ◽  
Magdalena Pereda
Keyword(s):  
Likelihood Ratio ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Discrete Distributions ◽  
Stochastic Orders ◽  
Survival Functions ◽  
Discrete Random Variables ◽  
Parametric Families

In this paper we focus on providing sufficient conditions for some well-known stochastic orders in reliability but dealing with the discrete versions of them, filling a gap in the literature. In particular, we find conditions based on the unimodality of the likelihood ratio for the comparison in some stochastic orders of two discrete random variables. These results have interest in comparing discrete random variables because the sufficient conditions are easy to check when there are no closed expressions for the survival functions, which occurs in many cases. In addition, the results are applied to compare several parametric families of discrete distributions.

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