The problem of bandwidth selection for kernel density estimation at a point is considered. Asymptotic lower bounds are established for the relative rate of convergence of data-driven bandwidth selectors to their optimal values. It is noted that some existing methods of local bandwidth selection, using high order kernel functions, attain these rates. Nevertheless, a simulation study indicates that improved performance predicted by asymptotic theory may not occur in practice for sample sizes as large as 10⁴ or 10⁵. The paper finishes with a comparison of local and global bandwidth selection, observing that in some sense the local problem is the more difficult.