Cramer-von mises-type tests with applications to tests of independence for multivariate extreme-value distributions
Titel:
Cramer-von mises-type tests with applications to tests of independence for multivariate extreme-value distributions
Auteur:
Deheuvels, Paul Martynov, Guennady V.
Verschenen in:
Communications in statistics
Paginering:
Jaargang 25 (1996) nr. 4 pagina's 871-908
Jaar:
1996
Inhoud:
Most Cramer -von Mises-type statistics of interest converge in distribution to [image omitted] for some Gaussian process {ξ(t): t ε I}. We present a numerically efficient method for the evaluation of (ω2≤x). First [image omitted] is approximated by [image omitted] 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/m2) This allows us to approximate (ω2≤x) by(ω2,m≤x) which in turn is evaluated by inverting the characteristic function [image omitted] , where [image omitted] and Km = (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 [image omitted] we test the hypothesis that θ(t) = 1 by a Cramer-von Mises-type test statistic, converging weakly as [image omitted] where ξ(t) is a Gaussian process with covariance function [image omitted] We tabulate the distribution and critical values of ω2 with a precision of 10-5. Further applications are discussed, including Anderson-Darling and Cramer-von Mises-type tests of fit when parameters are estimated.
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