Classical Extreme Value Theory

Classical extreme value statistics is concerned with the distributional properties of the maximum of a number of independent and identically distributed (iid) random variables when the number of variables becomes large. A partial result was obtained by Fréchet (Annales de la Société Polonaise de Mathematique, Cracow 6:193–213, 1927), while Fisher and Tippett (Proc Camb Philosoph Soc 24:180–190, 1928) discovered that there are three types of possible limiting or asymptotic distributions, which are now contained in the Extremal Types Theorem, which is discussed in the next section. These three asymptotic distributions are typically referred to as the Gumbel, Fréchet, and Weibull distributions. It is also common practice to refer to them as Type I, Type II, and Type III, in the same order. Important contributions to this theory were later made by Gnedenko (Ann Math 44:423–453, 1943), Gumbel (Statistics of Extremes. New York, NY: Columbia University Press, 1958), and de Haan (On regular variation and its applications to the weak convergence of sample extremes. Tract, Mathematical Centre, Amsterdam, 1970).