On high-dimensional sign tests

Abstract

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 highdimensional 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.

A High-Dimensional Nonparametric Multivariate Test for Mean Vector

“…The proposed test can be used as a basic building block to develop nonparametric tests in other important settings such as testing for sparse alternative or testing a hypothesis on coefficients in high-dimensional factorial designs (Zhong and Chen, 2011). A spatial sign based test was proposed for sphericity when p = O ( n 2 ) in Zou et al (2014), and spatial sign tests were proposed for testing uniformity on the unit sphere and other related null hypotheses when p / n → c for some positive constant c in Paindaveinez and Verdebout (2013). The techniques related to sign tests have the potential to be used to develop the high-dimensional theory for other classical nonparametric multivariate testing procedures, such as those based on spatial sign ranks (e.g., Möttönen and Oja, 1995) and ranks (e.g., Hallin and Davy Paindaveine, 2006).…”

Testing uniformity on high-dimensional spheres against monotone rotationally symmetric alternatives

“…In particular, Zou et al (2014) recently considered the high-dimensional version of the Hallin and Paindaveine (2006) sign tests of sphericity, whereas an extension to the high-dimensional case of the location sign test from Chaudhuri (1992) and Möttönen and Oja (1995) was recently proposed in Wang, Peng and Li (2015). Considering (iii) in high dimensions is particularly appealing since for moderate-to-large p, sign tests show excellent (fixed-p) efficiency properties (see Paindaveine and Verdebout (2015) for details). Also, the concentration-of-measure phenomenon may make the restriction to signs virtually void as the dimension p increases.…”

“…For fixed p, the test is based on the null asymptotic χ 2 p distribution of R n . In the high-dimensional setup, Paindaveine and Verdebout (2015) obtained the following asymptotic normality result under the null.…”

Tests of Concentration for Low-Dimensional and High-Dimensional Directional Data

“…Such data cannot be analyzed via standard statistical techniques and require developing new appropriate methods. In this vein, tests of hypotheses for high-dimensional directional data have been recently proposed in [5], [6], [8], [15] and [18]. While [5], [6], [8] and [18] focussed on the null hypothesis of uniformity on high-dimensional unit spheres, [15] tackled the high-dimensional spherical location problem.…”