Modeling clusters of extreme values

Markovich, Natalia M.

doi:10.1007/s10687-013-0176-3

Cited by 25 publications

(28 citation statements)

References 22 publications

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“…Due to dependence such exceedances tend to occur in clusters. Such clusters of rare events and the asymptotic distributions of the cluster and inter-cluster sizes have been widely studied due to numerous applications, see Ancona-Navarrete and Tawn (2000), Beirlant et al (2004), Ferro and Segers (2003), Markovich (2014), Markovich (2016a), Robert (2009), Robert (2013), Roberts et al (2006), Robinson et al (2000) among others. There are three approaches in the cluster size study, namely, the blocks method, the runs method and the inter-exceedance times method.…”

Section: Introductionmentioning

confidence: 99%

“…non-ergodic sequences with θ = 0 is given in Theorem 4 by Doukhan et al (2015). Using achievements regarding the limit geometric-like distribution of T 1 (x ρn ) derived in (Theorem 2, Markovich (2014), Markovich (2016a)), where the (1 − ρ n )th quantile x ρn of {X n } is taken as u n , we derive in Section 2 a limit distribution of the first hitting time and its expectation that specifies (6). The achievements are similarly extended to the second hitting time, Section 3.…”

Section: Introductionmentioning

confidence: 99%

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Clustering and hitting times of threshold exceedances and applications

Markovich

2017

IJDATS

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Abstract:We investigate exceedances of the process over a sufficiently high threshold. The exceedances determine the risk of hazardous events like climate catastrophes, huge insurance claims, the loss and delay in telecommunication networks. Due to dependence such exceedances tend to occur in clusters. The cluster structure of social networks is caused by dependence (social relationships and interests) between nodes and possibly heavy-tailed distributions of the node degrees. A minimal time to reach a large node determines the first hitting time. We derive an asymptotically equivalent distribution and a limit expectation of the first hitting time to exceed the threshold un as the sample size n tends to infinity. The results can be extended to the second and, generally, to the kth (k > 2) hitting times. Applications in large-scale networks such as social, telecommunication and recommender systems are discussed.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Clustering and hitting times of threshold exceedances and applications

Markovich

2017

IJDATS

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“…The cluster of exceedances is roughly defined as the consecutive exceedances of {X n } over the threshold {u n } between two consecutive non-exceedances. For more rigorous definitions, see [4][5][6]. It is important to investigate stochastic nature of extremes since it allows us to collect statistics or opinions more effectively in the clustered (network sampling) process.…”

Section: Introductionmentioning

confidence: 99%

Distribution and dependence of extremes in network sampling processes

Avrachenkov

Markovich

Sreedharan

2015

Compu Social Networks

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We explore the dependence structure in the sampled sequence of complex networks. We consider randomized algorithms to sample the nodes and study extremal properties in any associated stationary sequence of characteristics of interest like node degrees, number of followers, or income of the nodes in online social networks, which satisfy two mixing conditions. Several useful extremes of the sampled sequence like the kth largest value, clusters of exceedances over a threshold, and first hitting time of a large value are investigated. We abstract the dependence and the statistics of extremes into a single parameter that appears in extreme value theory called extremal index (EI). In this work, we derive this parameter analytically and also estimate it empirically. We propose the use of EI as a parameter to compare different sampling procedures. As a specific example, degree correlations between neighboring nodes are studied in detail with three prominent random walks as sampling techniques.

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“…These improvements do not change the proof of Theorem 2 (Markovich 2014) apart of one item. In formula (43) (Markovich 2014) one can use…”

mentioning

confidence: 99%

“…Hence, we can replace 'ρ n (1 − ρ n ) (j −1)θ ' and 'q n (1 − q n ) (j −1)θ ' in Theorem 2 (Markovich 2014) by 'θ 2 ρ n (1 − ρ n ) (j −1)θ ' and 'θ 2 q n (1 − q n ) (j −1)θ ', respectively, without the requirement of the mixing condition D (x ρ n ) as in Markovich (2014).…”

mentioning

confidence: 99%