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Efficient simulation for dependent rare events with applications to extremes. (English) Zbl 1385.65014
Let \((X_1,\dots,X_n)\) be a random vector and \(M=\max_i\,X_i\). The authors are interested in the estimation of \(\alpha(\gamma)=P(M>\gamma)\) using estimators based on \(E(\gamma)=\sum_{i=1}^{n}I_{[X_i>\gamma]}\). Two estimates are proposed and their properties studied. Aside from that, the authors are also interested in the estimation of \(\beta_n(\gamma)=E\,\big[Y\,I_{[E(\gamma\geq n)]}\big]\), \(n=1,2,\dots,d,\) where \(Y\) is an arbitrary random variable. No assumptions about the dependence of the events \([X_i>\gamma]\) or dependence of these events and random variable \(Y\) are done. Properties of the suggested estimators are studies together with their efficiency. The numerical performance is illustrated on several nontrivial examples.

MSC:
65C60 Computational problems in statistics (MSC2010)
65C05 Monte Carlo methods
62G32 Statistics of extreme values; tail inference
Software:
GitHub; QRM; RareMaxima
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References:
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