A note on the test for complete independence in the high-dimensional setting

Abstract

In this work, we propose a testing methodology for assessing complete independence in both low- and high-dimensional scenarios. The test statistic is constructed from likelihood ratio statistics used to test the independence of pairs of variables and is bounded in the interval (0,1). The properties of the test are explored through simulations in terms of power and bias, considering three different covariance structures: compound symmetric, first-order autoregressive, and band-Toeplitz-type covariance matrices. Furthermore, we propose asymptotic approximations based on a single Gamma distribution or a mixture of two Gamma distributions that are computationally efficient and straightforward to implement in practical applications. Numerical studies are provided to illustrate the accuracy of the proposed asymptotic approximations when compared with other models. An application to a high-dimensional financial asset returns dataset is also provided.