Extreme Value Distributions

Extreme Value distributions arise as limiting distributions for maximums or minimums (extreme values) of a sample of independent, identically distributed random variables, as the sample size increases. Extreme Value Theory (EVT) is the theory of modelling and measuring events which occur with very small probability. This implies its usefulness in risk modelling as risky events per definition happen with low probability. Thus, these distributions are important in statistics. These models, along with the Generalized Extreme Value distribution, are widely used in risk management, finance, insurance, economics, hydrology, material sciences, telecommunications, and many other industries dealing with extreme events. The class of Extreme Value Distributions (EVD’s) essentially involves three types of extreme value distributions, types I, II and III, defined below. Definition 1 (Extreme Value Distributions for maxima). The following are the standard Extreme Value distribution functions: (i) Gumbel (type I): Λ(x) = exp{− exp(−x)}, x ∈ R;