Testing Independence Among a Large Number of High-Dimensional Random Vectors

Abstract

Capturing dependence among a large number of high-dimensional random vectors is a very important and challenging problem. By arranging n random vectors of length p in the form of a matrix, we develop a linear spectral statistic of the constructed matrix to test whether the n random vectors are independent or not. Specifically, the proposed statistic can also be applied to n random vectors, each of whose elements can be written as either a linear stationary process or a linear combination of independent random variables. The asymptotic distribution of the proposed test statistic is established for the case of as n → ∞. To avoid estimating the spectrum of each random vector, a modified test statistic, which is based on splitting the original n vectors into two equal parts and eliminating the term that contains the inner structure of each random vector or time series, is constructed. The facts that the limiting distribution is normal and there is no need to know the inner structure of each investigated random vector result in simple implementation of the constructed test statistic. Simulation results demonstrate that the proposed test is powerful against several commonly used dependence structures. An empirical application to detecting dependence of the closed prices from several stocks in the S&P500 also illustrates the applicability and effectiveness of our provided test. Supplementary materials for this article are available online.

KEY WORDS:

• Central limit theorem
• Covariance stationary time series
• Empirical spectral distribution
• Independence test
• Large-dimensional sample covariance matrix
• Linear spectral statistics