Phase transition of Schott's statistic for high-dimensional heavy-tailed data

Abstract

Consider Schott's statistic (Schott, 2005) defined as the squared Frobenius norm of the sample correlation matrix for data from $\alpha$-regularly varying populations. We investigate its asymptotic distribution in a general framework characterized by data dimension p, sample size n, and regularly varying coefficients $\alpha$. In particular, we identify a phase transition phenomenon in the asymptotic behavior. For light-tailed populations ($\alpha > 3$), we revisit the $\alpha$-free asymptotic distribution but relax the constraint on the ratio of $p/n$. For heavy-tailed populations ($\alpha < 3$), we derive a new asymptotic normal distribution whose variance explicitly depends on $\alpha$. We also propose a consistent estimator for the asymptotic variance such that the standardized Schott's test statistic remains applicable for unknown location parameters and all $\alpha > 0$.