Tail dependence between order statistics

In this work, we introduce the s,ks,k-extremal coefficients   for studying the tail dependence between the ss-th lower and kk-th upper order statistics of a normalized random vector. If its margins have tail dependence then so do their order statistics, with the strength of bivariate tail dependence decreasing as two order statistics become farther apart. Some general properties are derived for these dependence measures which can be expressed via copulas of random vectors. Its relations with other extremal dependence measures used in the literature are discussed, such as multivariate tail dependence coefficients, the coefficient ηη of tail dependence, coefficients based on tail dependence functions, the extremal coefficient ϵϵ, the multivariate extremal index and an extremal coefficient for min-stable distributions. Several examples are presented to illustrate the results, including multivariate exponential and multivariate Gumbel distributions widely used in applications.