Cramer-von mises-type tests with applications to tests of independence for multivariate extreme-value distributions
Most Cramer -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 am and ti 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 ϵm = ω2- ω2,m satisfies Var(ϵm)=O(1/m 2) This allows us to approximate ℙ(ω2≤x) byℙ(ω2,m≤x) which in turn is evaluated by inverting the characteristic function , where and K m = (K(ti,tj ),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 Cramer-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 ...