{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T09:15:28Z","timestamp":1790846128826,"version":"4.1.0"},"reference-count":35,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2015,4,7]],"date-time":"2015-04-07T00:00:00Z","timestamp":1428364800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1208354"],"award-info":[{"award-number":["1208354"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,1,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>We derive non-parametric confidence intervals for the eigenvalues of the Hessian at modes of a density estimate. This provides information about the strength and shape of modes and can also be used as a significance test. We use a data-splitting approach in which potential modes are identified by using the first half of the data and inference is done with the second half of the data. To obtain valid confidence sets for the eigenvalues, we use a bootstrap based on an elementary symmetric polynomial transformation. This leads to valid bootstrap confidence sets regardless of any multiplicities in the eigenvalues. We also suggest a new method for bandwidth selection, namely choosing the bandwidth to maximize the number of significant modes. We show by example that this method works well. Even when the true distribution is singular, and hence does not have a density (in which case cross-validation chooses a zero bandwidth), our method chooses a reasonable bandwidth.<\/jats:p>","DOI":"10.1111\/rssb.12111","type":"journal-article","created":{"date-parts":[[2015,4,7]],"date-time":"2015-04-07T02:19:08Z","timestamp":1428373148000},"page":"99-126","source":"Crossref","is-referenced-by-count":33,"title":["Non-Parametric Inference for Density Modes"],"prefix":"10.1093","volume":"78","author":[{"given":"Christopher R.","family":"Genovese","sequence":"first","affiliation":[{"name":"Carnegie Mellon University , Pittsburgh , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marco","family":"Perone-Pacifico","sequence":"additional","affiliation":[{"name":"University of Rome , Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Isabella","family":"Verdinelli","sequence":"additional","affiliation":[{"name":"Carnegie Mellon University , Pittsburgh , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Larry","family":"Wasserman","sequence":"additional","affiliation":[{"name":"Carnegie Mellon University , Pittsburgh , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2015,4,7]]},"reference":[{"key":"2023021709315899900_rssb12111-cit-0001","volume-title":"On the estimation of the gradient lines of a density and the consistency of the mean-shift algorithm","author":"Arias-Castro","year":"2013"},{"key":"2023021709315899900_rssb12111-cit-0002","volume-title":"Feature selection for high-dimensional clustering","author":"Azizyan","year":"2014"},{"key":"2023021709315899900_rssb12111-cit-0003","volume-title":"Statistical inference for persistent homology","author":"Balakrishnan","year":"2013"},{"key":"2023021709315899900_rssb12111-cit-0004","first-page":"95","article-title":"Bootstrap tests and confidence regions for functions of a covariance matrix","author":"Beran","year":"1985","journal-title":"Ann. 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