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



is the Threshold (location) parameter


is the Scale parameter


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.