Optimality of the covariance-free estimator’s convergence rate

Determine whether the n^{-1/4} error rate obtained for the covariance-free risk estimator is optimal, or establish a sharper rate for the discrepancy between the covariance-free and covariance-dependent estimators.

Background

The covariance-free estimator replaces the population correction matrix with an observable triangular-system solution. Theorem 3.2 gives an n^{-1/4} bound for the difference between the covariance-free and covariance-dependent estimators, whereas the covariance-dependent estimator itself has an n^{-1/2} guarantee. The authors explicitly indicate that the n^{-1/4} rate may not be optimal, leaving open the problem of determining its optimality or improving it.