Extreme Value (Gumbel) Distribution for Quantile-Quantile Plots

The extreme value (Type I) distribution has the probability density function:

f(x) = 1/b * e-(x-a)/b * e**[-e-(x-a)/b]

-∞ < x < ∞ b > 0

where

a is the Threshold (location) parameter
b is the Scale parameter
e is the base of the natural logarithm, sometimes called Euler's e (2.71...)
  • The inverse distribution function (of probability a) is: (a-b)*loglog(1/a)
  • The standardized extreme value distribution function is used to determine the best fitting distribution.

In general if the points in the Q-Q plot form a straight line, then the respective family of distributions (Extreme Value distribution) provides a good fit to the data. In that case, the intercept and slope of the fitted line can be interpreted as graphical estimates of the threshold (a) and scale (b) parameters, respectively.