We introduce an L2-type test for testing mutual independence and banded dependence structure for high dimensional data. The test is constructed on the basis of the pairwise distance covariance and it accounts for the non-linear and non-monotone dependences among the data, which cannot be fully captured by the existing tests based on either Pearson correlation or rank correlation. Our test can be conveniently implemented in practice as the limiting null distribution of the test statistic is shown to be standard normal. It exhibits excellent finite sample performance in our simulation studies even when the sample size is small albeit the dimension is high and is shown to identify non-linear dependence in empirical data analysis successfully. On the theory side, asymptotic normality of our test statistic is shown under quite mild moment assumptions and with little restriction on the growth rate of the dimension as a function of sample size. As a demonstration of good power properties for our distance-covariance-based test, we further show that an infeasible version of our test statistic has the rate optimality in the class of Gaussian distributions with equal correlation.
- Award ID(s):
- 1916014
- NSF-PAR ID:
- 10355249
- Date Published:
- Journal Name:
- Journal of statistical computation and simulation
- ISSN:
- 0094-9655
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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null (Ed.)Summary This paper is concerned with empirical likelihood inference on the population mean when the dimension $p$ and the sample size $n$ satisfy $p/n\rightarrow c\in [1,\infty)$. As shown in Tsao (2004), the empirical likelihood method fails with high probability when $p/n>1/2$ because the convex hull of the $n$ observations in $\mathbb{R}^p$ becomes too small to cover the true mean value. Moreover, when $p> n$, the sample covariance matrix becomes singular, and this results in the breakdown of the first sandwich approximation for the log empirical likelihood ratio. To deal with these two challenges, we propose a new strategy of adding two artificial data points to the observed data. We establish the asymptotic normality of the proposed empirical likelihood ratio test. The proposed test statistic does not involve the inverse of the sample covariance matrix. Furthermore, its form is explicit, so the test can easily be carried out with low computational cost. Our numerical comparison shows that the proposed test outperforms some existing tests for high-dimensional mean vectors in terms of power. We also illustrate the proposed procedure with an empirical analysis of stock data.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1080/00949655.2022.2029860
