Cramer-von mises-type tests with applications to tests of independence for multivariate extreme-value distributions

Abstract

Most Cramér -von Mises-type statistics of interest converge in distribution to for some Gaussian process {ξ(t): t ϵ I}. We present a numerically efficient method for the evaluation of ℙ(ω²≤x). First is approximated by where a_m and t_i are the coefficients and nodes of a quadrature formula. Under suitable regularity conditions imposed upon the covariance function K of ξ, we show that the approximation error ϵ_m = ω²- ω^2,m satisfies Var(ϵ_m)=O(1/m ²) This allows us to approximate ℙ(ω²≤x) byℙ(ω^2,m≤x) which in turn is evaluated by inverting the characteristic function , where and K ^m = (K(t_i,t_j ),1 ≤ i,j ≤m) An application of this method is provided for tests of marginal independence for multivariate extreme values. Given a sample of size n from U = (U₁,U₂ ) with distribution function we test the hypothesis that θ(t) = 1 by a Cramér-von Mises-type test statistic, converging weakly as where ξ(t) is a Gaussian process with covariance function We tabulate the distribution and critical values of ω² with a precision of 10^-5. Further applications are discussed, including Anderson-Darling and Cramér-von Mises-type tests of fit when parameters are estimated.

Keywords:

• Extremes values
• multivariate distributions
• Cramér von Mises tests of fit
• tests of independence statistical tables