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Attributing a probability to the shape of a probability density

Hall, Peter; Ooi, Hong

Description

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...[Show more]

dc.contributor.authorHall, Peter
dc.contributor.authorOoi, Hong
dc.date.accessioned2016-03-03T22:52:03Z
dc.date.available2016-03-03T22:52:03Z
dc.identifier.issn0090-5364
dc.identifier.urihttp://hdl.handle.net/1885/99987
dc.description.abstractWe 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 Muller 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.
dc.publisherInstitute of Mathematical Statistics
dc.rights© Institute of Mathematical Statistics, 2004. http://www.sherpa.ac.uk/romeo/issn/0090-5364..."author can archive publisher's version/PDF. On author's personal website or open access repository" from SHERPA/RoMEO site (as at 4/03/16).
dc.sourceAnnals of Statistics
dc.subjectKeywords: Bandwidth choice; Bootstrap; Bump; Constrained inference; Data sharpening; Density estimation; Excess mass; Kernel methods; Likelihood; Mode; Nonparametric curve estimation; Subsampling
dc.titleAttributing a probability to the shape of a probability density
dc.typeJournal article
local.description.notesImported from ARIES
local.description.refereedYes
local.identifier.citationvolume32
dc.date.issued2004
local.identifier.absfor010405
local.identifier.ariespublicationMigratedxPub9937
local.publisher.urlhttp://imstat.org/en/index.html
local.type.statusPublished Version
local.contributor.affiliationHall, Peter, College of Physical and Mathematical Sciences, CPMS Mathematical Sciences Institute, Centre for Mathematics and Its Applications, The Australian National University
local.contributor.affiliationOoi, Hong, College of Physical and Mathematical Sciences, CPMS Mathematical Sciences Institute, Centre for Mathematics and Its Applications, The Australian National University
local.bibliographicCitation.issue5
local.bibliographicCitation.startpage2098
local.bibliographicCitation.lastpage2123
local.identifier.doi10.1214/009053604000000607
dc.date.updated2016-06-14T08:37:35Z
local.identifier.scopusID2-s2.0-24344488636
dcterms.accessRightsOpen Access
CollectionsANU Research Publications

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