The Annals of Statistics

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

Joseph P. Romano

Full-text: Open access


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

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

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

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


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.

Export citation