It is well known that data-driven regression smoothing parametersħbased on cross-validation and related methods exhibit a slow rate of convergence to their optimum. In an earlier article we showed that this rate can be as slow asn–1/10; that is, for a bandwidthħ0optimizing the averaged squared error,n1/10(ħ—ħ0)/ħ0tends to an asymptotic normal distribution. In this article we consider mean averaged squared error optimal bandwidthsh0. This (nonrandom) smoothing parameter can be approximated much faster. We use the technique of double smoothing to show that there is anħsuch that, under certain conditions,n1/2(ħ−h0)/h0tends to an asymptotic normal distribution.