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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
Full Text:
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