On multivariate runs tests for randomness

This paper proposes several extensions of the concept of runs to the multivariate setup, and studies the resulting tests of multivariate randomness against serial dependence. Two types of multivariate runs are defined: (i) an elliptical extension of the spherical runs proposed by Marden (1999), and (ii) an original concept of matrix-valued runs. The resulting runs tests themselves exist in various versions, one of which is a function of the number of data-based hyperplanes separating pairs of observations only. All proposed multivariate runs tests are affine-invariant and highly robust: in particular, they allow for heteroscedasticity and do not require any moment assumption. Their limiting distributions are derived under the null hypothesis and under sequences of local vector ARMA alternatives. Asymptotic relative efficiencies with respect to Gaussian Portmanteau tests are computed, and show that, while Marden-type runs tests suffer severe consistency problems, tests based on matrix-valued runs perform uniformly well for moderate-to-large dimensions. A Monte Carlo study confirms the theoretical results and investigates the robustness properties of the proposed procedures. A real-data example is also treated, and shows that combining both types of runs tests may provide some insight on the reason why rejection occurs, hence that Marden-type runs tests, despite their lack of consistency, also are of interest for practical purposes. KEY WORDS: Elliptical distributions; Interdircctions; Local asymptotic normality; Multivariate signs; Shape matrix.