Sign tests are among the most successful procedures in multivariate nonparametric statistics. In this paper, we consider several testing problems in multivariate analysis, directional statistics and multivariate time series analysis, and we show that, under appropriate symmetry assumptions, the fixed-$p$ multivariate sign tests remain valid in the high-dimensional case. Remarkably, our asymptotic results are universal, in the sense that, unlike in most previous works in high-dimensional statistics, $p$ may go to infinity in an arbitrary way as $n$ does. We conduct simulations that (i) confirm our asymptotic results, (ii) reveal that, even for relatively large $p$, chi-square critical values are to be favoured over the (asymptotically equivalent) Gaussian ones and (iii) show that, for testing i.i.d.-ness against serial dependence in the high-dimensional case, Portmanteau sign tests outperform their competitors in terms of validity-robustness.
"On high-dimensional sign tests." Bernoulli 22 (3) 1745 - 1769, August 2016. https://doi.org/10.3150/15-BEJ710