The Annals of Statistics

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

Bernard W. Silverman

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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.

Article information

Source
Ann. Statist., Volume 6, Number 1 (1978), 177-184.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176344076

Digital Object Identifier
doi:10.1214/aos/1176344076

Mathematical Reviews number (MathSciNet)
MR471166

Zentralblatt MATH identifier
0376.62024

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 41A25: Rate of convergence, degree of approximation 60F15: Strong theorems 60G15: Gaussian processes 60G17: Sample path properties

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

Citation

Silverman, Bernard W. Weak and Strong Uniform Consistency of the Kernel Estimate of a Density and its Derivatives. Ann. Statist. 6 (1978), no. 1, 177--184. doi:10.1214/aos/1176344076. https://projecteuclid.org/euclid.aos/1176344076


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See also

  • Addendum: Bernard W. Silverman. Addendum to Weak and Strong Uniform Consistency of the Kernel Estimate of a Density and Its Derivatives. Ann. Statist., Volume 8, Number 5 (1980), 1175--1176.