In this paper, we consider a financial or insurance system with a finite number of individual risks described by real-valued random variables. We focus on two kinds of risk measures, referred to as the tail moment (TM) and the tail central moment (TCM), which are defined as the conditional moment and conditional central moment of some individual risk in the event of system crisis. The first-order TM and the second-order TCM coincide with the popular risk measures called the marginal expected shortfall and the tail variance, respectively. We derive asymptotic expressions for the TM and TCM with any positive integer orders, when the individual risks are pairwise asymptotically independent and have distributions from certain classes that contain both light-tailed and heavy-tailed distributions. The formulas obtained possess concise forms unrelated to dependence structures, and hence enable us to estimate the TM and TCM efficiently. To demonstrate the wide application of our results, we revisit some issues related to premium principles and optimal capital allocation from the asymptotic point of view. We also give a numerical study on the relative errors of the asymptotic results obtained, under some specific scenarios when there are two individual risks in the system. The corresponding asymptotic properties of the degenerate univariate versions of the TM and TCM are discussed separately in an appendix at the end of the paper.
{"title":"Asymptotic results on tail moment and tail central moment for dependent risks","authors":"Jinzhu Li","doi":"10.1017/apr.2022.74","DOIUrl":"https://doi.org/10.1017/apr.2022.74","url":null,"abstract":"\u0000 In this paper, we consider a financial or insurance system with a finite number of individual risks described by real-valued random variables. We focus on two kinds of risk measures, referred to as the tail moment (TM) and the tail central moment (TCM), which are defined as the conditional moment and conditional central moment of some individual risk in the event of system crisis. The first-order TM and the second-order TCM coincide with the popular risk measures called the marginal expected shortfall and the tail variance, respectively. We derive asymptotic expressions for the TM and TCM with any positive integer orders, when the individual risks are pairwise asymptotically independent and have distributions from certain classes that contain both light-tailed and heavy-tailed distributions. The formulas obtained possess concise forms unrelated to dependence structures, and hence enable us to estimate the TM and TCM efficiently. To demonstrate the wide application of our results, we revisit some issues related to premium principles and optimal capital allocation from the asymptotic point of view. We also give a numerical study on the relative errors of the asymptotic results obtained, under some specific scenarios when there are two individual risks in the system. The corresponding asymptotic properties of the degenerate univariate versions of the TM and TCM are discussed separately in an appendix at the end of the paper.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44091059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the literature on active redundancy allocation, the redundancy lifetimes are usually postulated to be independent of the component lifetimes for the sake of technical convenience. However, this unrealistic assumption leads to a risk of inaccurately evaluating system reliability, because it overlooks the statistical dependence of lifetimes due to common stresses. In this study, for k-out-of-n:F systems with component and redundancy lifetimes linked by the Archimedean copula, we show that (i) allocating more homogeneous redundancies to the less reliable components tends to produce a redundant system with stochastically larger lifetime, (ii) the reliability of the redundant system can be uniformly maximized through balancing the allocation of homogeneous redundancies in the context of homogeneous components, and (iii) allocating a more reliable matched redundancy to a less reliable component produces a more reliable system. These novel results on k-out-of-n:F systems in which component and redundancy lifetimes are statistically dependent are more applicable to the complicated engineering systems that arise in real practice. Some numerical examples are also presented to illustrate these findings.
{"title":"Stochastic comparison on active redundancy allocation to k-out-of-n systems with statistically dependent component and redundancy lifetimes","authors":"Yinping You, Xiaohu Li, Xiaoqin Li","doi":"10.1017/apr.2022.70","DOIUrl":"https://doi.org/10.1017/apr.2022.70","url":null,"abstract":"\u0000 In the literature on active redundancy allocation, the redundancy lifetimes are usually postulated to be independent of the component lifetimes for the sake of technical convenience. However, this unrealistic assumption leads to a risk of inaccurately evaluating system reliability, because it overlooks the statistical dependence of lifetimes due to common stresses. In this study, for k-out-of-n:F systems with component and redundancy lifetimes linked by the Archimedean copula, we show that (i) allocating more homogeneous redundancies to the less reliable components tends to produce a redundant system with stochastically larger lifetime, (ii) the reliability of the redundant system can be uniformly maximized through balancing the allocation of homogeneous redundancies in the context of homogeneous components, and (iii) allocating a more reliable matched redundancy to a less reliable component produces a more reliable system. These novel results on k-out-of-n:F systems in which component and redundancy lifetimes are statistically dependent are more applicable to the complicated engineering systems that arise in real practice. Some numerical examples are also presented to illustrate these findings.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43527329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study (asymmetric) $U$ -statistics based on a stationary sequence of $m$ -dependent variables; moreover, we consider constrained $U$ -statistics, where the defining multiple sum only includes terms satisfying some restrictions on the gaps between indices. Results include a law of large numbers and a central limit theorem, together with results on rate of convergence, moment convergence, functional convergence, and a renewal theory version. Special attention is paid to degenerate cases where, after the standard normalization, the asymptotic variance vanishes; in these cases non-normal limits occur after a different normalization. The results are motivated by applications to pattern matching in random strings and permutations. We obtain both new results and new proofs of old results.
{"title":"Asymptotic normality for \u0000$boldsymbol{m}$\u0000 -dependent and constrained \u0000$boldsymbol{U}$\u0000 -statistics, with applications to pattern matching in random strings and permutations","authors":"S. Janson","doi":"10.1017/apr.2022.51","DOIUrl":"https://doi.org/10.1017/apr.2022.51","url":null,"abstract":"Abstract We study (asymmetric) \u0000$U$\u0000 -statistics based on a stationary sequence of \u0000$m$\u0000 -dependent variables; moreover, we consider constrained \u0000$U$\u0000 -statistics, where the defining multiple sum only includes terms satisfying some restrictions on the gaps between indices. Results include a law of large numbers and a central limit theorem, together with results on rate of convergence, moment convergence, functional convergence, and a renewal theory version. Special attention is paid to degenerate cases where, after the standard normalization, the asymptotic variance vanishes; in these cases non-normal limits occur after a different normalization. The results are motivated by applications to pattern matching in random strings and permutations. We obtain both new results and new proofs of old results.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"55 1","pages":"841 - 894"},"PeriodicalIF":1.2,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45373391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gabriel Zayas-Cabán, Jiaxin Liang, Stefanus Jasin, Guihua Wang
The above lemma is used to prove Theorems 1–2 and Propositions 1–3 in Sections 4 and 6 of [1]. It has been graciously pointed out to us that the bound in the lemma may not be correct in general. The original proof of this lemma uses a combination of linear program (LP) duality and sensitivity analysis results. The mistake is in the application of a known sensitivity analysis result under a certain assumption that happens to be not necessarily satisfied by our LP. Fortunately, it is possible to correct the bound in the above lemma. The new bound that we will prove in this correction note is as follows:
{"title":"An asymptotically optimal heuristic for general nonstationary finite-horizon restless multi-armed, multi-action bandits: Corrigendum","authors":"Gabriel Zayas-Cabán, Jiaxin Liang, Stefanus Jasin, Guihua Wang","doi":"10.1017/apr.2022.59","DOIUrl":"https://doi.org/10.1017/apr.2022.59","url":null,"abstract":"The above lemma is used to prove Theorems 1–2 and Propositions 1–3 in Sections 4 and 6 of [1]. It has been graciously pointed out to us that the bound in the lemma may not be correct in general. The original proof of this lemma uses a combination of linear program (LP) duality and sensitivity analysis results. The mistake is in the application of a known sensitivity analysis result under a certain assumption that happens to be not necessarily satisfied by our LP. Fortunately, it is possible to correct the bound in the above lemma. The new bound that we will prove in this correction note is as follows:","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"55 1","pages":"1075 - 1083"},"PeriodicalIF":1.2,"publicationDate":"2023-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46883874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract During an epidemic outbreak, typically only partial information about the outbreak is known. A common scenario is that the infection times of individuals are unknown, but individuals, on displaying symptoms, are identified as infectious and removed from the population. We study the distribution of the number of infectives given only the times of removals in a Markovian susceptible–infectious–removed (SIR) epidemic. Primary interest is in the initial stages of the epidemic process, where a branching (birth–death) process approximation is applicable. We show that the number of individuals alive in a time-inhomogeneous birth–death process at time $t geq 0$ , given only death times up to and including time t, is a mixture of negative binomial distributions, with the number of mixing components depending on the total number of deaths, and the mixing weights depending upon the inter-arrival times of the deaths. We further consider the extension to the case where some deaths are unobserved. We also discuss the application of the results to control measures and statistical inference.
{"title":"The size of a Markovian SIR epidemic given only removal data","authors":"F. Ball, P. Neal","doi":"10.1017/apr.2022.58","DOIUrl":"https://doi.org/10.1017/apr.2022.58","url":null,"abstract":"Abstract During an epidemic outbreak, typically only partial information about the outbreak is known. A common scenario is that the infection times of individuals are unknown, but individuals, on displaying symptoms, are identified as infectious and removed from the population. We study the distribution of the number of infectives given only the times of removals in a Markovian susceptible–infectious–removed (SIR) epidemic. Primary interest is in the initial stages of the epidemic process, where a branching (birth–death) process approximation is applicable. We show that the number of individuals alive in a time-inhomogeneous birth–death process at time \u0000$t geq 0$\u0000 , given only death times up to and including time t, is a mixture of negative binomial distributions, with the number of mixing components depending on the total number of deaths, and the mixing weights depending upon the inter-arrival times of the deaths. We further consider the extension to the case where some deaths are unobserved. We also discuss the application of the results to control measures and statistical inference.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"55 1","pages":"895 - 926"},"PeriodicalIF":1.2,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41387185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Consider Bernoulli bond percolation on a graph nicely embedded in hyperbolic space $mathbb{H}^d$ in such a way that it admits a transitive action by isometries of $mathbb{H}^d$ . Let $p_{text{a}}$ be the supremum of all percolation parameters such that no point at infinity of $mathbb{H}^d$ lies in the boundary of the cluster of a fixed vertex with positive probability. Then for any parameter $p < p_{text{a}}$ , almost surely every percolation cluster is thin-ended, i.e. has only one-point boundaries of ends.
{"title":"Thin-ended clusters in percolation in \u0000$mathbb{H}^d$","authors":"J. Czajkowski","doi":"10.1017/apr.2022.43","DOIUrl":"https://doi.org/10.1017/apr.2022.43","url":null,"abstract":"Abstract Consider Bernoulli bond percolation on a graph nicely embedded in hyperbolic space \u0000$mathbb{H}^d$\u0000 in such a way that it admits a transitive action by isometries of \u0000$mathbb{H}^d$\u0000 . Let \u0000$p_{text{a}}$\u0000 be the supremum of all percolation parameters such that no point at infinity of \u0000$mathbb{H}^d$\u0000 lies in the boundary of the cluster of a fixed vertex with positive probability. Then for any parameter \u0000$p < p_{text{a}}$\u0000 , almost surely every percolation cluster is thin-ended, i.e. has only one-point boundaries of ends.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"55 1","pages":"581 - 610"},"PeriodicalIF":1.2,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49120931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：