Very fast optimal bandwidth selection for univariate kernel density estimation

Most automatic bandwidth selection procedures for kernel density estimates require estimation of quantities involving the density derivatives. Estimation of modes and in∞exion points of densities also require derivative estimates. The computational complexity of evaluating the density derivative at M evaluation points given N sample points from the density is O(MN). In this paper we propose a computationally e‐cient †iexact approximation algorithm for the univariate Gaussian kernel based density derivative estimation that reduces the computational complexity from O(MN) to linear O(N + M). The constant depends on the desired arbitrary accuracy, †. We apply the density derivative evaluation procedure to estimate the optimal bandwidth for kernel density estimation, a process that is often intractable for large data sets. For example for N = M = 409;600 points while the direct evaluation of the density derivative takes around 12:76 hours the fast evaluation requires only 65 seconds with an error of around 10 i12 . Algorithm details, error bounds, procedure to choose the parameters and numerical experiments are presented. We demonstrate the speedup achieved on the bandwidth selection using the solve-the-equation plug-in method. We also demonstrate that the proposed procedure can be extremely useful for speeding up exploratory projection pursuit techniques.