AISM 51, 541-569

© 1999 ISM

## Counting bumps

### Ricardo Fraiman^{1} and Jean Meloche^{2}

^{1}Centro de Matemática, Universidad de la República, Eduardo Acevedo 1139 Montevideo, 11200, Uruguay and Universidad de San Andrés, Buenos Aires, Argentina

^{2}Department of Statistics, University of British Columbia, 6356 Agricultural Road, Vancouver, BC, Canada V6T 1Z2

(Received September 27, 1996; revised December 15, 1997)

Abstract.
The number of modes of a density $f$ can be estimated by counting the number of 0-downcrossings of an estimate of the derivative $f'$, but this often results in an overestimate because random fluctuations of the estimate in the neighbourhood of points where $f$ is nearly constant will
induce spurious counts. Instead of counting the number of 0-downcrossings, we count the number of "significant" modes by counting the number of downcrossings of an interval $[-\epsilon,\epsilon]$. We obtain consistent estimates and confidence intervals for the number of "significant" modes.
By letting $\epsilon$ converge slowly to zero, we get consistent estimates of the number of modes. The same approach can be used to estimate the number of critical points of any derivative of a density function, and in particular the number of inflection points.

Key words and phrases:
Significant bumps, density estimation, downcrossings, confidence intervals, bandwidth selection.

**Source**
(TeX , DVI , PS)