Logarithmic law of large random correlation matrix

Abstract

Consider a random vector y = Σ 1/2 x, where the p elements of the vector x are i.i.d. real-valued random variables with zero mean and finite fourth moment, and Σ 1/2 is a deterministic p × p matrix such that the spectral norm of the population correlation matrix R of y is uniformly bounded. In this paper, we find that the log determinant of the sample correlation matrix R based on a sample of size n from the distribution of y satisfies a CLT (central limit theorem) for p/n → γ ∈ (0, 1] and p ≤ n. Explicit formulas for the asymptotic mean and variance are provided. In case the mean of y is unknown, we show that after recentering by the empirical mean the obtained CLT holds with a shift in the asymptotic mean. This result is of independent interest in both large dimensional random matrix theory and high-dimensional statistical literature of large sample correlation matrices for non-normal data. At last, the obtained findings are applied for testing of uncorrelatedness of p random variables. Surprisingly, in the null case R = I, the test statistic becomes completely pivotal and the extensive simulations show that the obtained CLT also holds if the moments of order four do not exist at all, which conjectures a promising and robust test statistic for heavy-tailed high-dimensional data.

Likelihood ratio tests under model misspecification in high dimensions

“…, y n . Several authors investigated tests for the hypothesis given in (5) in different frameworks (e.g., see Jiang and Yang (2013), Jiang and Qi (2015), Gao et al (2017), Mestre and Vallet (2017), Qi et al (2019), Parolya et al (2021), Heiny and Parolya (2021)). We observe that testing for (5) is a special case of testing for (2) by letting q = p and p 1 = .…”