The Annals of Statistics

On Weak Convergence and Optimality of Kernel Density Estimates of the Mode

Joseph P. Romano

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Abstract

A mode of a probability density $f(t)$ is a value $\theta$ that maximizes $f$. The problem of estimating the location of the mode is considered here. Estimates of the mode are considered via kernel density estimates. Previous results on this problem have several serious drawbacks. Conditions on the underlying density $f$ are imposed globally (rather than locally in a neighborhood of $\theta$). Moreover, fixed bandwidth sequences are considered, resulting in an estimate of the location of the mode that is not scale-equivariant. In addition, an optimal choice of bandwidth depends on the underlying density, and so cannot be realized by a fixed bandwidth sequence. Here, fixed and random bandwidths are considered, while relatively weak assumptions are imposed on the underlying density. Asymptotic minimax risk lower bounds are obtained for estimators of the mode and kernel density estimates of the mode are shown to possess a certain optimal local asymptotic minimax risk property. Bootstrapping the sampling distribution of the estimates is also discussed.

Article information

Source
Ann. Statist., Volume 16, Number 2 (1988), 629-647.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176350824

Digital Object Identifier
doi:10.1214/aos/1176350824

Mathematical Reviews number (MathSciNet)
MR947566

Zentralblatt MATH identifier
0658.62053

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62E20: Asymptotic distribution theory 62G20: Asymptotic properties

Keywords
Kernel density estimates mode weak convergence rates of convergence asymptotic minimax risk

Citation

Romano, Joseph P. On Weak Convergence and Optimality of Kernel Density Estimates of the Mode. Ann. Statist. 16 (1988), no. 2, 629--647. doi:10.1214/aos/1176350824. https://projecteuclid.org/euclid.aos/1176350824


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