One of the major problems in kernel density estimation is the choice of bandwidth. We review the first order properties of the likelihood cross-validation bandwidth selection method, introduced by Habbema, Hermans and Van den Broek and Duin. This method was modified by Marron 1985 to obtain an asymptotic optimality property for Lipschitz densities. In particular we show that for densities with jumps the modified method loses this optimality property. Furthermore we establish the asymptotic normality of the distance of the likelihood cross-validation bandwidth to the minimizer of a suitable integrated squared error. It turns out that for certain non smooth densities this distance has a faster rate of convergence to zero than for smooth densities.