  It is shown that the variable bandwidth density estimator proposed by McKay
(1993a and b) following earlier findings by Abramson (1982) approximates
density functions in $C^4(\mathbb R^d)$ at the minimax rate in the supremum
norm over bounded sets where the preliminary density estimates on which they
are based are bounded away from zero. A somewhat more complicated estimator
proposed by Jones McKay and Hu (1994) to approximate densities in $C^6(\mathbb
R)$ is also shown to attain minimax rates in sup norm over the same kind of
sets. These estimators are strict probability densities.
