{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,11]],"date-time":"2026-08-11T20:57:23Z","timestamp":1786481843741,"version":"3.56.0"},"reference-count":18,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,9,1]],"date-time":"2021-09-01T00:00:00Z","timestamp":1630454400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["21H00699 and 21K17713"],"award-info":[{"award-number":["21H00699 and 21K17713"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Although robust divergence, such as density power divergence and \u03b3-divergence, is helpful for robust statistical inference in the presence of outliers, the tuning parameter that controls the degree of robustness is chosen in a rule-of-thumb, which may lead to an inefficient inference. We here propose a selection criterion based on an asymptotic approximation of the Hyvarinen score applied to an unnormalized model defined by robust divergence. The proposed selection criterion only requires first and second-order partial derivatives of an assumed density function with respect to observations, which can be easily computed regardless of the number of parameters. We demonstrate the usefulness of the proposed method via numerical studies using normal distributions and regularized linear regression.<\/jats:p>","DOI":"10.3390\/e23091147","type":"journal-article","created":{"date-parts":[[2021,9,1]],"date-time":"2021-09-01T11:39:03Z","timestamp":1630496343000},"page":"1147","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["On Selection Criteria for the Tuning Parameter in Robust Divergence"],"prefix":"10.3390","volume":"23","author":[{"given":"Shonosuke","family":"Sugasawa","sequence":"first","affiliation":[{"name":"Center for Spatial Information Science, The University of Tokyo, Chiba 277-8568, Japan"},{"name":"Nospare Inc., Tokyo 107-0061, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shouto","family":"Yonekura","sequence":"additional","affiliation":[{"name":"Nospare Inc., Tokyo 107-0061, Japan"},{"name":"Graduate School of Social Sciences, Chiba University, Chiba 263-8522, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1093\/biomet\/85.3.549","article-title":"Robust and efficient estimation by minimising a density power divergence","volume":"85","author":"Basu","year":"1998","journal-title":"Biometrika"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2053","DOI":"10.1016\/j.jmva.2008.02.004","article-title":"Robust parameter estimation with a small bias against heavy contamination","volume":"99","author":"Fujisawa","year":"2008","journal-title":"J. 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