Summary: Kernel smoothers belong to the most popular nonparametric functional estimates. These smoothers depend on three parameters: the bandwidth which controls the smoothness of the estimate, the form of the kernel weight function and the order of the kernel which is related to the number of derivatives assumed to exist in the nonparametric model. Because these three problems are closely related one to each other it is necessary to address them all together.
We concentrate on the estimation of a density function and of its derivatives. We propose to use polynomial kernels and we construct data-driven choices for the bandwidth and the order of the kernel. We prove a theorem stating that this method for solving simultaneously the three selection problems mentioned before is asymptotically optimal in terms of Mean Integrated Squared Errors. As a by-product of our result, we show an asymptotic optimality property for a new bandwidth selector for density derivatives which is quite appealing because of the simplicity of its implementation. As another by-product, we extend the notion of canonical kernel to the setting of derivative estimation.