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October 2004 Attributing a probability to the shape of a probability density
Peter Hall, Hong Ooi
Ann. Statist. 32(5): 2098-2123 (October 2004). DOI: 10.1214/009053604000000607


We discuss properties of two methods for ascribing probabilities to the shape of a probability distribution. One is based on the idea of counting the number of modes of a bootstrap version of a standard kernel density estimator. We argue that the simplest form of that method suffers from the same difficulties that inhibit level accuracy of Silverman’s bandwidth-based test for modality: the conditional distribution of the bootstrap form of a density estimator is not a good approximation to the actual distribution of the estimator. This difficulty is less pronounced if the density estimator is oversmoothed, but the problem of selecting the extent of oversmoothing is inherently difficult. It is shown that the optimal bandwidth, in the sense of producing optimally high sensitivity, depends on the widths of putative bumps in the unknown density and is exactly as difficult to determine as those bumps are to detect. We also develop a second approach to ascribing a probability to shape, using Müller and Sawitzki’s notion of excess mass. In contrast to the context just discussed, it is shown that the bootstrap distribution of empirical excess mass is a relatively good approximation to its true distribution. This leads to empirical approximations to the likelihoods of different levels of “modal sharpness,” or “delineation,” of modes of a density. The technique is illustrated numerically.


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Peter Hall. Hong Ooi. "Attributing a probability to the shape of a probability density." Ann. Statist. 32 (5) 2098 - 2123, October 2004.


Published: October 2004
First available in Project Euclid: 27 October 2004

zbMATH: 1057.62028
MathSciNet: MR2102504
Digital Object Identifier: 10.1214/009053604000000607

Primary: 62G07
Secondary: 62G20

Keywords: Bandwidth choice , bootstrap , bump , constrained inference , data sharpening , Density estimation , excess mass , kernel methods , likelihood , Mode , Nonparametric curve estimation , subsampling

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 5 • October 2004
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