Counting bumps

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.