An on-line density estimator may be defined to be one where each update, following the arrival of a new data value, may be accomplished after no more than a fixed number of calculations. This definition should also apply to any empirical bandwidth selection rule for such an estimator. Recursive estimators comprise only a special case of on-line estimators, but even there, on-line bandwidth formulas have not been developed. The authors introduce a class of on-line estimators, and discuss efficiency in this context. It is shown that some nonrecursive members of the class achieve greater efficiency than any recursive estimators, and that efficiency increases to 100% as the order of the estimated derivative increases. On-line bandwidth selection rules, enabling these high orders of efficiency to be achieved asymptotically, are introduced.>