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 [-∈, ∈]. We obtain consistent estimates and confidence intervals for the number of "significant" modes. By letting ∈ 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.
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Fraiman, R., Meloche, J. Counting Bumps. Annals of the Institute of Statistical Mathematics 51, 541–569 (1999). https://doi.org/10.1023/A:1003958323950
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DOI: https://doi.org/10.1023/A:1003958323950