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January, 1978 Weak and Strong Uniform Consistency of the Kernel Estimate of a Density and its Derivatives
Bernard W. Silverman
Ann. Statist. 6(1): 177-184 (January, 1978). DOI: 10.1214/aos/1176344076


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.


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Bernard W. Silverman. "Weak and Strong Uniform Consistency of the Kernel Estimate of a Density and its Derivatives." Ann. Statist. 6 (1) 177 - 184, January, 1978.


Published: January, 1978
First available in Project Euclid: 12 April 2007

zbMATH: 0376.62024
MathSciNet: MR471166
Digital Object Identifier: 10.1214/aos/1176344076

Primary: 62G05
Secondary: 41A25 , 60F15 , 60G15 , 60G17

Keywords: density derivative estimate , density estimate , Gaussian process , global consistency , ‎kernel‎ , modulus of continuity , rates of convergence , supremum over real line

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 1 • January, 1978
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