Over-threshold sampling : Modeling of occurrences by renewal processes
The principle of over-threshold sampling is to consider all the events in a time-series that exceed a given threshold. The probabilistic analysis implies estimating two statistical models, one describing the occurrence of events (date of the events), the other describing their magnitude (value of the local maximum). These two models are then combined to obtain the distribution of annual maximum flows. The theory of renewal processes can be used to study the occurrence of flood events. We present here properties of the well-known Poisson distribution (stationary or non-stationary process), and certain new results for the binomial and negative binomial distributions. These results concern the distribution of the number of events in a given time interval and the distribution of the waiting time, defined as the time span between two successive exceedances of the threshold. The relationship between the distribution of a variable and its corresponding return period are then studied in more detail. We review the results for the case where the variable of interest is obtained by selection of one or more events per year, or from a Poisson point process. New results are presented for the case of the binomial and negative binomial processes. Finally, we establish the analytical relationship between the two types of sampling, annual maximum sampling and peaks-over-threshold sampling, in terms of return period, distribution of the annual maximum, and sampling variance.