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I'm interested in estimating the distributions of a few skewed datasets, for example extreme heat, and extreme rainfall.

There are many distributions that can be fit to these kinds of data, for instance, this page shows an attempt to fit multiple distributions to flood data, including Generalised Extreme Value, Generalised Pareto, Wakeby, Lognormal, and Gumbel distributions.

The fits look like this:

Generalised Extreme Value, Generalised Pareto, Wakeby, Lognormal, and Gumbel distributions fit to flood data

The page decides to use the Generalised Pareto distribution, based purely on statistical performance. There are a few problems with this that I can see: First, I can't see the underlying data, so it's possible they are fitting on only a small handful of data, and also possible that the coarseness of the histogram is hiding interesting features of the data (e.g. a down-turn near zero). They don't provide uncertainty estimates on the performance stats either, so it's not clear if the other distributions' confidence intervals overlap. It's also possible that the sample they're fitting is biased somehow, and so skewing the result towards the a particular distribution.

With the data that I'm fitting, I'm seeing quite similar results between Generalised Extreme Value and Log Normal - at some locations, the GEV fit looks better, at others, the Log Normal fit appears to do better. However, they are all always the same kind of data, and so I feel like I should always use a consistent model. As such, I would like to know:

Is there ever a purely theoretical/conceptual basis for choosing one distribution over another, assuming that the distributions perform similarly?

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First I'd like to preface by saying that fitting distributions to complicated real world data doesn't really help as much as one would think. Distributions are often generated through well-defined statistical processes, and unless your data comes from a regular and well-behaved physical or mathematical process, the identity of its distribution isn't going to give you much insight.

In regards to your actual question, which I consider part of curve/model fitting in general, you should always go for the simplest model/distribution that fits your data well. By 'simple', I mean one with the fewest parameters, but also one that the general science/math community is likely to be familiar with (I am familiar with lognormal, but not with Wakeby, for instance). Personally, my intuition is to try to fit the data above with an exponential decay.

Special care needs to be taken with extrapolation, however. The example is about flood frequency, and presumably one would want to know about the relative frequency of the big but rare floods. Many of the distributions in that plot, although they behave similarly in the interval displayed, they'll behave differently in the long tail. For instance, the gumbel max decays faster than the lognormal, which decays (I believe) faster than the generalised Pareto. Your choice of distribution affects the predicted rate of occurrence for the floods that are rarer than what's in your data.

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  • $\begingroup$ Your final paragraph is the crux of the question! If I'm fitting data with 30-50 annual maximums, then the tail is likely to be poorly represented. I would have thought that since distributions like the GEV are specifically designed for maxima estimation, then they'd be good at optimising for that tail. But I'm not sure why the log normal wouldn't be just as good, given that I don't have data to test it. I guess I could select some of the longer site datasets, and try some cross-validation with a tail-specific performance metric... But it would be nice if there were some theoretical guidance. $\endgroup$ – naught101 Oct 15 '18 at 3:04
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    $\begingroup$ Honestly, if I knew that, I wouldn't be wasting all my time on stackexchange. I suspect it's not just me either, and that guessing at the behaviour of the tail is one of those significant unsolved problems of statistics. Use of the GEV distribution presupposes your data also follows a well-understood distribution (a normal distribution I believe for Gumbel). In any case, choosing a model with few parameters still holds, as you have less danger of overfitting. The Wakeby distribution for instance has five parameters that could easily overfit and create strange behaviour in the tails. $\endgroup$ – Ingolifs Oct 15 '18 at 3:26
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    $\begingroup$ Makes sense. I guess I can be content with "I have no idea why I'm using GEV over log normal, but at least no one else does either" :D The wikipedia page says that Gumbel is good for normal or exponential distributions, so it should be OK for my data, I think. $\endgroup$ – naught101 Oct 15 '18 at 3:49

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