I am computing the Generalized Extreme Value distribution for a dataset containing about 15 years of data sampled every 5 seconds. I want to estimate the 1-in-50 or 1-in-100 year return level from the GEV. I originally used a block length of one year to compute the GEV (e.g. take the maximum of each 1 year interval and use that as input into the GEV fitting). This allowed me to easily compute the return probability of exceedance as:
$$P_{exceed} = 1-1/T_{return}$$
where $T_{return}$ was 50 (or 100) years. Then I could find the GEV level that has a probability of exceedance, $P_{exceed}$.
However, the dataset often has more than one independent "extreme value" in a given year, and could have as many as 10 in a given year, so taking an annual block maxima will underestimate the number of extreme events and the return levels.
How can I modify this in order to get the 1-in-50 year return levels while simultaneously using an e.g. 10-day block length in the GEV?
I thought I could just compute the GEV using the 10-day blocks and then multiply $T_{return}$ by 36.5 to get the annual $P_{exceed}$ rather than the 10-day $P_{exceed}$. This seems to work, but I am unsure if it is statistcally valid.
There a couple additional caveats here:
Firstly, my data has some gaps. If I do annual block maxima, then there is no problem because each block still has some data to compute the maxima, but if I do a 10-day block length, there are some blocks which have no data at all and thus have a block maxima of NaN. Is it better to just pretend the gaps don't exist and that the time series is shorter but continuous? (That's what I am currently doing.) Or is it better to assume the gaps have very small maxima and just give them some default small value?
Secondly, extreme events are rare (duh). If I do an annual block maxima, I am basically guaranteed to get an "extreme" event in every block. But if I do a 10-day block length, then a lot of those blocks have very small maxima. The GEV fits those smallest block maxima well, but fits the furthest extremes the worst (see figures below).
Any clarification is appreciated!
Thanks
Example #1: Annual block maxima. The CI are really large, but the best fitting GEV passes through the extremes really well at least 
Example #2: 10-day block maxima. The CI are much smaller, but the best fitting GEV does not even remotely pass through the extreme values after about 1-year return interval. Why is the fit so bad? 
