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

Abstract

Consider Schott&#39;s statistic (Schott, 2005) defined as the squared Frobenius norm of the sample correlation matrix for data from $α$-regularly varying populations. We investigate its asymptotic distribution in a general framework characterized by data dimension p, sample size n, and regularly varying coefficients $α$. In particular, we identify a phase transition phenomenon in the asymptotic behavior. For light-tailed populations ($α> 3$), we revisit the $α$-free asymptotic distribution but relax the constraint on the ratio of $p/n$. For heavy-tailed populations ($α< 3$), we derive a new asymptotic normal distribution whose variance explicitly depends on $α$. We also propose a consistent estimator for the asymptotic variance such that the standardized Schott&#39;s test statistic remains applicable for unknown location parameters and all $α> 0$.