On some kernel density estimation bandwidth selectors related to the double kernel method

We explore an L2 modification of an L1-based method for automatic band width selection for kernel density estimation called the double kernel method (Devroye, 1989). The method uses up to two bandwidths, g and h, in objective function estimation. When g = h, we observe close relationships with a particular useful class of objective functions that contains two existing bandwidth selection methods as special cases, and are led to links with higher order kernels and bias correction. When g >> h, the L2 double kernel method does not provide the expected improvement in quality of estimating the mean integrated squared error (MISE) optimal bandwidth. Instead, it stays close to its roots in estimating the ISE optimal bandwidth and provides a particularly good estimator thereof