Statistics of Weibull Record-Breaking Events

Abstract

The statistics of record-breaking events plays an important role in the analysis of natural physical systems. It can provide an additional insight into the mechanisms and the occurrence of extreme events. In this work, the statistical aspects of the record-breaking events drawn from the Weibull distribution are considered and analyzed in detail. It is assumed that the underlying sequences of events are independent and identically distributed (i.i.d.). Several statistical measures of record-breaking events are analyzed. Exact analytical expressions are derived for the statistics of records. Particularly, the distributions of record magnitudes and the corresponding average magnitudes of records in case of Weibull distributed events are derived exactly for any specific record order and time step. In addition, a convolution operation is used to derive a recursive formula for the distribution of times of the occurrence of records. The analytical results are compared with the Monte Carlo simulations and their validity is confirmed. The numerical simulations also reveal that the finite-size effects strongly affect the statistics of records and need to be considered during the analysis of numerical experiments or empirical data.