Extreme values are the largest or the smallest observations in a sample; e.g., the sample minimum (the first order statistic) and the sample maximum (the n-th order statistic). Associated with extreme values are asymptotic *extreme value distributions.*
Extreme values are the largest or the smallest observations in the sample, e.g., sample minimum (the first order statistics), sample maximum (the n-th order statistic), the second smallest/largest values (the 2nd and the (n-1)st order statistic, respectively), etc. Extreme values are often associated with outliers or catastrophic events, and have application in modeling floods in climatology, value-at-risk in finance, etc.
With a simple negation, the problems concerning the minima and the smallest values can be converted into problems concerning the maxima and the largest values. Hence, the extreme value theory results are typically formulated in terms of the right tail and maxima.
The central result in the extreme value theory concerns the asymptotic distribution of the maximum (the Fisher–Tippett–Gnedenko theorem). In large samples, an appropriately scaled sample maximum follows one of three possible distributional families: Gumbel, Frechet or Weibull distributions. The common functional form for the three is sometimes referred to as the generalized extreme value distribution. Another theorem in extreme value theory is the Pickands-Balkema-de Haan theorem. It concerns the asymptotic distribution of values above a certain threshold. These peak over threshold values can be well approximated by the generalized pareto distribution. Other results of extreme value distribution theory concern the spacings; i.e., the distance from the largest to the second largest value, the second largest to the third largest value, etc.
Related tags: maximum, minimum
