Sharp minimax theory for Sobolev covariance classes with diagonal bias

Derive a sharp minimax theory for covariance estimation over Sobolev classes with integer regularity, accounting for the diagonal-band contribution to the integrated squared bias, and determine whether diagonal-corrected smoothers recover the h^{2\beta} bias rate.

Background

The paper notes that Fourier-decay covariance classes with integer regularity can fail to be Hölder classes of the same order because of a logarithmic singularity along the diagonal. For these Sobolev-type covariances, the integrated squared bias is observed to scale as h^{2\beta-1}, rather than h^{2\beta}.

The diagonal band therefore determines the bias behavior and may alter minimax rates even though the threshold exponent associated with symmetry is unchanged. A complete minimax treatment and the potential correction of this effect by specialized smoothers remain unresolved.