## The Annals of Statistics

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

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

#### 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

#### 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.Project Euclid: euclid.aos/1176345157