The Annals of Statistics

Nonparametric testing of the existence of modes

Michael C. Minnotte

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Given a set of data drawn from an unknown density, it is frequently desirable to estimate the number and location of modes of the density. A test is proposed for the weight of evidence of individual observed modes. The test statistic used is a measure of the size of the mode, the absolute integrated difference between the estimated density and the same density with the mode in question excised at the level of the higher of its two surrounding antimodes. Samples are simulated from a conservative member of the composite null hypothesis to estimate p-values within a Monte Carlo setting. Such a test can be used with the graphical "mode tree" of Minnotte and Scott to examine, in a locally adaptive fashion, not only the reality of individual modes, but also (roughly) the overall number of modes of the density. A proof of consistency of the test statistic is offered and simulation results are presented.

Article information

Ann. Statist. Volume 25, Number 4 (1997), 1646-1660.

First available in Project Euclid: 9 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing 62G07: Density estimation
Secondary: 62G09: Resampling methods 62G20: Asymptotic properties

Bump hunting kernel density estimation mode estimation multi-modality


Minnotte, Michael C. Nonparametric testing of the existence of modes. Ann. Statist. 25 (1997), no. 4, 1646--1660. doi:10.1214/aos/1031594735.

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