Asymptotic linearity of serial and nonserial multivariate signed rank statistics

Asymptotic linearity plays a key role in estimation and testing in the presence of nuisance parameters. This property is established, in the very general context of a multivariate general linear model with elliptical VARMA errors, for the serial and nonserial multivariate rank statistics considered in Hallin and Paindaveine (Ann. Statist. 30 (2002a) 1103; Bernoulli 8 (2002b) 787 Ann. Statist. 32 (2004), to appear) and Oja and Paindaveine (J. Statist. Plann. Inference (2004), to appear). These statistics, which are multivariate versions of classical signed rank statistics, involve (i) multivariate signs based either on (pseudo-)Mahalanobis residuals, or on a modified version (absolute interdirections) of Randles's interdirections, and (ii) a concept of ranks based either on (pseudo-)Mahalanobis distances or on lift-interdirections.