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June, 1988 On Weak Convergence and Optimality of Kernel Density Estimates of the Mode
Joseph P. Romano
Ann. Statist. 16(2): 629-647 (June, 1988). DOI: 10.1214/aos/1176350824


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.


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Joseph P. Romano. "On Weak Convergence and Optimality of Kernel Density Estimates of the Mode." Ann. Statist. 16 (2) 629 - 647, June, 1988.


Published: June, 1988
First available in Project Euclid: 12 April 2007

zbMATH: 0658.62053
MathSciNet: MR947566
Digital Object Identifier: 10.1214/aos/1176350824

Primary: 62G05
Secondary: 62E20, 62G20

Rights: Copyright © 1988 Institute of Mathematical Statistics


Vol.16 • No. 2 • June, 1988
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