The selection of the order, i.e., number of vanishing moments, of the kernel in a kernel density estimator is considered from two points of view. First, theoretical properties are investigated by a mean integrated squared error analysis of the problem. Second, and perhaps more importantly, cross validation is proposed as a practical method of choice, and theoretical backing for this is provided through an asymptotic optimality result.
"Choice of Kernel Order in Density Estimation." Ann. Statist. 16 (1) 161 - 173, March, 1988. https://doi.org/10.1214/aos/1176350697