Generalized Gumbel Probability Distribution Function

Computes the Generalized Gumbel PDF for modeling data with a broad range of skewness

The Generalized Gumbel (GG) PDF is the generalization of the standard Gumbel PDF (i.e., Type 1 Extreme Value PDF https://www.mathworks.com/help/stats/evpdf.html). In the GG PDF, the skewness parameter "a" (related to the third moment of the distribution) can quantify a range of skewness in the data. When "a" = 1, the GG PDF reduces to the standard Gumbel PDF with an exponential tail. When "a" > 1, the GG PDF becomes increasingly Gaussian; the parameter "a" diverges for a Gaussian PDF.

This simple code calculates the GG PDFs based on input data x, wherein x is normalized according to x_norm = (x-mu)/std. The code can easily be tweaked to calculate GG PDFs for the unnormalized data x, or to serve as the basis of a regression-based fitting function for noisy data.

Functional forms for the GG PDFs are taken from Planet et al., Phys Rev Lett 2009 (positive skew) and
Barucci et al., Euro Phys Lett 2010 (negative skew). Additional information on the usage of the GG PDF for modeling critical phenomena can be found in Bertin et al., Phys Rev Lett 2005.