  We show that scale-adjusted versions of the centroid-based classifier enjoys
optimal properties when used to discriminate between two very high-dimensional
populations where the principal differences are in location. The scale
adjustment removes the tendency of scale differences to confound differences in
means. Certain other distance-based methods, for example, those founded on
nearest-neighbor distance, do not have optimal performance in the sense that we
propose. Our results permit varying degrees of sparsity and signal strength to
be treated, and require only mild conditions on dependence of vector
components. Additionally, we permit the marginal distributions of vector
components to vary extensively. In addition to providing theory we explore
numerical properties of a centroid-based classifier, and show that these
features reflect theoretical accounts of performance.
