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Maximum likelihood fits aren't very robust with GEV distributions. It would be nice to implement the method of moments as an alternative.

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It would be good to have a function that implements a Hill estimator, for example. The Hill estimator is particularly useful in extreme value theory for analyzing the tail properties of a distribution. It focuses on the most extreme observations (the largest) in a sample to estimate the tail index, which informs about the heaviness of the tail.

MLE is not always converging to the ground truth:

using Distributions
using ExtremeStats

# make sure example is reproducible
srand(2018)

# sample data from a known distribution
true_gev = GeneralizedExtremeValue(1.,2.,0.1)
xs = rand(true_gev, 100000)

# MLE on block maxima
bm = BlockMaxima(xs, 1000)
mle_gev = fit(GeneralizedExtremeValue, bm)

display(true_gev)
display(mle_gev)

Distributions.GeneralizedExtremeValue{Float64}(μ=1.0, σ=2.0, ξ=0.1)
Distributions.GeneralizedExtremeValue{Float64}(μ=20.77381198318063, σ=3.4459184571273136, ξ=1.5852450471032418e-6)

My understanding is that it should converge to a set of parameters that is at least close. The parameters obtained are far off.

We should migrate to Makie recipes and fix the visual test infrastructure.