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It just occurred to me to ask this.

It's easy to get a statistical distribution that does not have finite variance. For example, you can sometimes get that when the thing that is measured comes from two random variables, one divided by the other. Sometimes you can't measure a variance. Sometimes you can get a statistical distribution that doesn't even have a mean.

When that happens, if you don't notice, you can get a mean and standard distribution from the data. And when you collect more data it will seem to mostly fit. But you get big outliers more often than you'd expect. As you recompute your mean and standard deviation, with more data the standard deviation keeps increasing. Because the longer you keep measuring, the more unexpected events you will have that go outside the predicted range.

Without knowing much at all about floods, they seem to fit this pattern. The news keeps announcing floods that were supposed to be unlikely, unexpected.

Of course, for all I know this is just the news reporting things that should have been expected. Of all the thousands of places we calculate hundred-year-flood levels for, every year we should expect floods that high at one percent of them. Maybe what's happening is really exactly what should be expected.

But it's testable. With enough data you can check whether flooding ought to fit a finite-variance distribution or not.

How well has it been tested?

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  • $\begingroup$ If they would have zero variance, their recurrence would be exactly periodical. Is that really the question you have? $\endgroup$
    – AtmosphericPrisonEscape
    Commented Dec 25, 2018 at 15:03
  • $\begingroup$ No, that wasn't the question. I'm interested in how MUCH variability they have. People like to assume normal distributions because lots of things tend to average out to that. But some things don't. Sometimes they average out to something with more outliers than you'd expect. That's worth paying attention to. Rivercfd explained that the statistical model that gets used in fact is one that does not assume finite variance. I'm trying to look at the implications of that, in my limited free time. $\endgroup$
    – J Thomas
    Commented Dec 25, 2018 at 15:34
  • $\begingroup$ The sample distribution will have a finite variance, correct? So I'm not sure how it is really testable. In a practical sense, there is an implicit assumption that the mechanisms like climate are not changing with time so you may not be sampling from the same distribution. Ultimately, you need to have a distribution that is explainable by the underlying physical processes. For example, we know that the maximum flood is limited by the maximum precipitation. $\endgroup$
    – haresfur
    Commented Dec 26, 2018 at 22:09
  • $\begingroup$ Yes, any sample distribution will have finite variance. But some mathematical models generate distributions that do not have finite variance. So even if you keep sampling from the same distribution, you will get outliers more frequently than you might expect. You can get distributions with fat tails. $\endgroup$
    – J Thomas
    Commented Dec 27, 2018 at 7:40

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I'm sure a statistics guru over on cross-validated could answer this better for you, but here's some general info on estimating flood frequency fyi.

The 100 year flood has a 1% probability of occurring in any given year. It is typically estimated by fitting some type of an extreme value distribution to observed flood peaks with additional skew adjustments to account for regional influences or upstream regulation. In the US, the standard approach is to fit a Log Pearson Type III distribution to log transformed observed flood peaks. Until recently, Bulletin#17B was the FEMA approved method for flood risk hydrology when a sufficient record of flood peaks was available. It was superseded March 2018 by Bulletin#17C which has some improved statistical methods.

As far as your question regarding finite variance, my understanding is that the LP III distribution asymptotically approaches unity and thus remains unbounded without finite variance since from a risk perspective there could always be a larger flood that has not yet been measured. There are also methods for estimating a Probable Maximum Flood (PMF) that is meant to represent the most extreme combination of meteorological and hydrologic conditions that are reasonably possible in a catchment.

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