Pickands–Balkema–de Haan theorem

The Pickands–Balkema–de Haan theorem is often called the second theorem in extreme value theory. It gives the asymptotic tail distribution of a random variable X, when the true distribution F of X is unknown. Unlike for the first theorem (the Fisher–Tippett–Gnedenko theorem) in extreme value theory, the interest here is in the values above a threshold. The Pickands–Balkema–de Haan theorem is often called the second theorem in extreme value theory. It gives the asymptotic tail distribution of a random variable X, when the true distribution F of X is unknown. Unlike for the first theorem (the Fisher–Tippett–Gnedenko theorem) in extreme value theory, the interest here is in the values above a threshold. If we consider an unknown distribution function F {displaystyle F} of a random variable X {displaystyle X} , we are interested in estimating the conditional distribution function F u {displaystyle F_{u}} of the variable X {displaystyle X} above a certain threshold u {displaystyle u} . This is the so-called conditional excess distribution function, defined as for 0 ≤ y ≤ x F − u {displaystyle 0leq yleq x_{F}-u} , where x F {displaystyle x_{F}} is either the finite or infinite right endpoint of the underlying distribution F {displaystyle F} . The function F u {displaystyle F_{u}} describes the distribution of the excess value over a threshold u {displaystyle u} , given that the threshold is exceeded. Let ( X 1 , X 2 , … ) {displaystyle (X_{1},X_{2},ldots )} be a sequence of independent and identically-distributed random variables, and let F u {displaystyle F_{u}} be their conditional excess distribution function. Pickands (1975), Balkema and de Haan (1974) posed that for a large class of underlying distribution functions F {displaystyle F} , and large u {displaystyle u} , F u {displaystyle F_{u}} is well approximated by the generalized Pareto distribution. That is: