Asymptotic conditional distribution of exceedance counts. (English) Zbl 1236.62005

Falk, Michael; Tichy, Diana

Adv. Appl. Probab. 44, No. 1, 270-291 (2012).

Summary: We investigate the asymptotic distribution of the number of exceedances among \(d\) identically distributed but not necessarily independent random variables (RVs) above a sequence of increasing thresholds, conditional on the assumption that there is at least one exceedance. Our results enable the computation of the fragility index, which represents the expected number of exceedances, given that there is at least one exceedance. Computed from the first \(d\) RVs of a strictly stationary sequence, we show that, under appropriate conditions, the reciprocal of the fragility index converges to the extremal index corresponding to the stationary sequence as \(d\) increases.

MSC:

62E20	Asymptotic distribution theory in statistics
62G32	Statistics of extreme values; tail inference
60G70	Extreme value theory; extremal stochastic processes

Keywords:

exceedance over high threshold; fragility index; multivariate extreme value theory; peaks-over-threshold approach; copula; generalized Pareto distribution (GPD); GPD copula; D-norm; extremal index