skewness is for ξ>0 For ξ<0, the sign of the numerator is reversed. The excess kurtosis is: where , , and is the gamma function. Link to Fréchet, Weibull and Gumbel families The shape parameter governs the tail behavior of the distribution. The sub-families defined by , and correspond, respectively, to the Gumbel, Fréchet and Weibull families, whose cumulative distribution functions are displayed below. Gumbel or type I extreme value distribution () Fréchet or type II extreme value distribution, if and Reversed Weibull or type III extreme value distribution, if and The subsections below remark on properties of these distributions. Modification for minima rather than maxima The theory here relates to data maxima and the distribution being discussed is an extreme value distribution for maxima. A generalised extreme value distribution for data minima can be obtained, for example by substituting (−x) for x in the distribution function, and subtracting from one: this yields a separate family of distributions. Alternative convention for the Weibull distribution The ordinary Weibull distribution arises in reliability applications and is obtained from the distribution here by using the variable , which gives a strictly positive support - in contrast to the use in the extreme value theory here. This arises because the ordinary Weibull distribution is used in cases that deal with data minima rather than data maxima. The distribution here has an addition parameter compared to the usual form of the Weibull distribution and, in addition, is reversed so that the distribution has an upper bound rather than a lower bound. Importantly, in applications of the GEV, the upper bound is unknown and so must be estimated, while when applying the ordinary Weibull distribution in reliability applications the lower bound is usually known to be zero. Ranges of the distributions Note the differences in the ranges of interest for the three extreme value distributions: Gumbel is unlimited, Fréchet has a lower limit, while the reversed Weibull has an upper limit. More precisely, Extreme Value Theory (Univariate Theory) describes which of the three is the limiting law according to the initial law X and in particular depending on its tail. Distribution of log variables One can link the type I to types II and III in the following way: if the cumulative distribution function of some random variable is of type II, and with the positive numbers as support, i.e. , then the cumulative distribution function