Multivariate sign-based high-dimensional tests for sphericity

Abstract

5This article concerns tests for sphericity when data dimension is larger than the sample size. The existing multivariate-sign-based procedure (Hallin & Paindaveine, 2006) for sphericity is not robust against high dimensionality, producing tests with type I error rates much larger than nominal levels. This is mainly due to bias from estimating the location parameter. We develop a correction that makes the existing test statistic robust against high dimensionality. We show that the proposed test statistic is asymp-10 totically normal under elliptical distributions. The proposed method allows dimensionality to increase as the square of sample size. Simulations show that it has good size and power for a wide range of settings.

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 the equality of two high‐dimensional spatial sign covariance matrices

“…For simplicity, we let n = n 1 + n 2 . Similar to Zou, Peng, Feng, and Wang (2014), Assumption 1 is a necessary condition to ensure the validity of the second-order expansions. Assumption 2 is similar to Li and Chen (2012).…”

Multivariate-Sign-Based High-Dimensional Tests for the Two-Sample Location Problem

“…However, this would yield a bias-term which is not negligible (with respect to the standard deviation) when n/p = O(1) because we replace the true spatial median with its estimate. It seems infeasible to develop a bias-correction procedure as done in Zou et al (2014) because the bias term depends on the unknown quantities Σ i 's. Please refer to Appendix C in the Supplemental Material for the closed-form of this bias and associated asymptotic analysis.…”

“…Like the Hotelling's T 2 test mentioned above, traditional methods may not work any more in this situation since they assume that p keeps unchanged as n increases. This challenge calls for new statistical tests to deal with highdimensional data, see Dempster (1958), Bai and Saranadasa (1996), Srivastava and Du (2008) and Chen and Qin (2010) for two-sample tests for means, Ledoit and Wolf (2002), Schott (2005 and Zou et al (2014) for testing a specific covariance structure, Goeman et al (2006), Zhong and Chen (2011) and Feng et al (2013) for high-dimensional regression coefficients.…”

“…First, we restate the Lemma 4 inZou et al (2014) here.18ACCEPTED MANUSCRIPTDownloaded by [New York University] at 06:03 12 July 2015 ACCEPTED MANUSCRIPT Lemma 1 Suppose u are independent identically distributed uniform on the unit p sphere. For any p × p symmetric matrix M, we have E(u T Mu) 2 ={tr 2 (M) + 2tr(M 2 )}/(p 2 + 2p), E(u T Mu) 4 ={3tr 2 (M 2 ) + 6tr(M 4 )}/{p(p + 2)(p + 4)(p + 6)}.…”