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A multivariate Wald-Wolfowitz rank test against serial dependence. (English) Zbl 0821.62022
Summary: Rank-based cross-covariance matrices, extending to the case of multivariate observed series the (univariate) rank autocorrelation coefficients introduced by A. Wald and J. Wolfowitz [Ann. Math. Stat. 14, 378-388 (1943; Zbl 0060.302)], are considered. A permutational central limit theorem is established for the joint distribution of such matrices, under the null hypothesis of (multivariate) randomness as well as under contiguous alternatives of (multivariate) ARMA dependence. A rank-based, permutationallly distribution-free test of the portmanteau type is derived, and its asymptotic local power is investigated. Finally, a modified rank-based version of G. C. Tiao and G. E. P. Box’s [J. Am. Stat. Assoc. 76, 802-816 (1981; Zbl 0483.62074)] model specification procedure is proposed, which is likely to be more reliable under non-Gaussian conditions, and more robust against gross errors.

MSC:
62G10 Nonparametric hypothesis testing
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62H15 Hypothesis testing in multivariate analysis
62E20 Asymptotic distribution theory in statistics
62H10 Multivariate distribution of statistics
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