Weak and Strong Uniform Consistency of the Kernel Estimate of a Density and its Derivatives

TLDR: In this article, the estimation of a density and its derivatives by the kernel method is considered, and uniform consistency properties over the whole real line are studied under certain conditions on the density and on the behavior of the window width which are necessary and sufficient for weak and strong uniform consistency of the estimate of the density derivatives.

Abstract: The estimation of a density and its derivatives by the kernel method is considered. Uniform consistency properties over the whole real line are studied. For suitable kernels and uniformly continuous densities it is shown that the conditions $h \rightarrow 0$ and $(nh)^{-1} \log n \rightarrow 0$ are sufficient for strong uniform consistency of the density estimate, where $n$ is the sample size and $h$ is the "window width." Under certain conditions on the kernel, conditions are found on the density and on the behavior of the window width which are necessary and sufficient for weak and strong uniform consistency of the estimate of the density derivatives. Theorems on the rate of strong and weak consistency are also proved.