# How to assess the quality of a return period estimate?

Background I have a time series of structural loads, which are measured forces on a moored ocean buoy, and I need to obtain the return period value so that the structure can be designed to withstand the max load expected in a storm. The return period is estimated from the measured time series in the following way: First, I choose a threshold and identify the peaks above this threshold. Extreme events are expected to have a distribution known as the Generalized Pareto Distribution (GPD), so I fit the peaks to that distribution. Then the GPD fit is inverted to give the load associated with the probability of a chosen time period (3 hours). This load is called the return period value. I am using a program called WAFO, which takes extreme value analysis routines from S-Plus, to do this analysis.

Problem The problem is how to assess the quality of the return period estimate. WAFO produces quantile-quantile plots that show how well the peaks match the GPD, a p-value to check the quality of the fit, or confidence bounds on the return period estimate. But these 3 diagnostics seem to conflict. Sometimes the q-q plot shows that many peaks don't fit the GPD (bad), but the p-value for the fit is high (good) and the confidence bounds are narrow (good). Sometimes the confidence bounds are very wide (bad) but the q-q plot shows the peaks lining up with the GPD (good). My gut sense is that the quality might be poor because we only measured data for a limited duration and there is likely measurement noise.

Question: How can I tell if the return period estimate is good vs bad?

• Just bought 'An introduction to statistical modeling of extreme values' by Stuart Coles, which looks like it will help. He recommends calculating the CI by the profile likelihood method rather than the delta method as I have been doing, and also notes that the CI is not assessing the quality of the fit - that's what the plots are for. Will post any insights after reading further.
– KAE
Commented Jan 15, 2013 at 14:16

## 1 Answer

Based on the Coles book, which I highly recommend, the best check on the goodness of fit is the quantile-quantile plot. The confidence interval does not address whether the model was correct or not. The CI should be seen as a lower bounds that would be much greater if the model correctness were considered (p.57 of Coles). You still need to look at it, it's just not the whole story. And for extreme values, the CI should be estimated using the profile likelihood method rather than the less accurate delta method that I was using previously. That was a big improvement.