I have a data set of peak-over-threshold values which I have fitted a generalised Pareto distribution to. From this, I wish to determine values corresponding to certain return periods in years. I have therefore been converting these return periods into t-year probabilities with the following:
$P_t = (1-(1/RP))^{(1/n)}$
where RP is the return period in years, $P_t$ is the t-year cumulative probability and n is the average number of events per year. RP is defined as follows:
$RP = 1/P_{ex}$
where $P_{ex}$ is the 1-year probability of exceedance and is equal to $1-P_1$, where $P_1$ is the 1-year cumulative probability.
If I wish to determine the probability of no exceedance for a given number of periods, lets say n periods, which equals 1-year, then I would use:
$P_1 = P_t^n$
Therefore:
$P_t = P_1^{1/n}$
$P_t = (1-P_{ex})^{1/n}$
$P_t = (1-(1/RP))^{(1/n)}$
However, I have found several papers which use the formula:
$P_t = 1-(1/(n*RP))$ (Rosbjerg, 1985) (Eq. 4)
And I can't seem to work out why this is different, and now I'm doubting my method. It seems in the paper they take the distributions of peaks in time as Poisson distribution, whereas I do not. Could this account for the difference in methods? Is there an issue with my own method? Any help is much appreciated.
Paper: Rosbjerg 1985, ESTIMATION IN PARTIAL DURATION SERIES WITH INDEPENDENT AND DEPENDENT PEAK VALUES