In the setting of kernel density estimation, data-driven bandwidth, i.e., smoothing parameter, selectors are considered. It is seen that there is a well-defined, and surprisingly restrictive, bound on the rate of convergence of any automatic bandwidth selection method to the optimum. The method of least squares cross-validation achieves this bound.
"On the Amount of Noise Inherent in Bandwidth Selection for a Kernel Density Estimator." Ann. Statist. 15 (1) 163 - 181, March, 1987. https://doi.org/10.1214/aos/1176350259