Tests for high-dimensional data based on means, spatial signs and spatial ranks

Abstract

Tests based on sample mean vectors and sample spatial signs have been studied in the recent literature for high dimensional data with the dimension larger than the sample size. For suitable sequences of alternatives, we show that the powers of the mean based tests and the tests based on spatial signs and ranks tend to be same as the data dimension grows to infinity for any sample size, when the coordinate variables satisfy appropriate mixing conditions. Further, their limiting powers do not depend on the heaviness of the tails of the distributions. This is in striking contrast to the asymptotic results obtained in the classical multivariate setup. On the other hand, we show that in the presence of stronger dependence among the coordinate variables, the spatial sign and rank based tests for high dimensional data can be asymptotically more powerful than the mean based tests if in addition to the data dimension, the sample size also grows to infinity. The sizes of some mean based tests for high dimensional data studied in the recent literature are observed to be significantly different from their nominal levels. This is due to the inadequacy of the asymptotic approximations used for the distributions of those test statistics. However, our asymptotic approximations for the tests based on spatial signs and ranks are observed to work well when the tests are applied on a variety of simulated and real datasets.

Mean Test with Fewer Observation than Dimension and Ratio Unbiased Estimator for Correlation Matrix

“…Other research related to sparsity can be seen from Zhong et al (2013), Chen et al (2014), Wang et al (2015), Gregory et al (2015) and Guo and Chen (2016). The researchers Biswas and Ghosh (2014), Chang et al (2017), Chakraborty and Chaudhuri (2017), and Xue and Yao (2020) consider the above test under non-normal assumptions. Other contributions include Park and Ayyala (2013) and Wu et al (2006) for the consideration of scale-invariant tests and Gretton et al (2012) for a kernel-based discrepancy measure.…”

Power in High‐Dimensional Testing Problems

“…For a discussion of several examples in the context of financial econometrics (testing implications from multi‐factor pricing theory) or panel data models (tests for cross‐sectional independence in mixed effect panels), refer to Fan, Liao, and Yao (). Testing problems in one‐ or two‐sample multivariate location models, that is, tests for the hypothesis whether the mean vector of a population is 0 or whether the mean vectors of two populations are identical, were analyzed in Dempster (), Bai and Saranadasa (), Srivastava and Du (), Srivastava, Katayama, and Kano (), Cai, Liu, and Xia (), and Chakraborty and Chaudhuri (); in this context, the articles by Pinelis (, ), where the asymptotic efficiency of tests based on different p ‐norms relative to the Euclidean norm were studied, need to be mentioned. For some concrete examples of high‐dimensional location models arising in empirical economics, see the discussion in Section 2.1 of Abadie and Kasy (2018).…”

Diagonal Likelihood Ratio Test for Equality of Mean Vectors in High-Dimensional Data

“…We also note that our current paper has focused on testing high-dimensional mean vectors under the parametric setting. More recently, some nonparametric tests have also been developed in the literature for the same testing problems; see, for example, Wang et al (2015), Ghosh and Biswas (2016), and Chakraborty and Chaudhuri (2017).…”