Probability Distribution of Waiting Time of the kth Extreme Event under Serial Dependence

Abstract

Negative binomial distribution has been suggested to describe the first arrival time of the kth flood exceeding the design flood under independence and both stationary and nonstationary conditions. However, hydrological processes often exhibit temporal dependence, which can cause persistent fluctuations in observed series and clustering of extreme events that might be confused with nonstationary effects. This study focuses on a distribution of waiting time of the kth event exceeding a prescribed design value under stationarity and serial dependence. This probability distribution is known as beta negative binomial, which complements the models proposed for (non)stationary independent processes, and enables the comparison with results corresponding to stationary dependent processes. We discuss the properties of the beta negative binomial distribution and show its validity for theoretical occurrence processes with power-law and exponentially decaying autocorrelation functions. The proposed model is applied to peak flows and maximum temperatures recorded across the conterminous United States. Results show that the beta negative binomial distribution can capture the effect of serial dependence on the distribution of waiting time of extreme events.