COUNTING BUMPS

RICARDO FRAIMAN^{1} AND JEAN MELOCHE^{2}

1Centro de Matemática, Universidad de la República, Eduardo Acevedo 1139,

Montevideo, 11200, Uruguay and Universidad de San Andrés, Buenos Aires, Argentina

2Department 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 [−ϵ, ϵ]. We obtain consistent estimates and confidence intervals for the number of “significant” modes. By letting ϵ 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.