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In particular, we propose the so-called <jats:italic>bi-orthogonal density estimator<\/jats:italic> based on the class of B-splines and derive its theoretical properties, including the asymptotically optimal choice of bandwidth. Detailed theoretical analysis and comparisons of our estimator with existing local basis and kernel density estimators are presented. The estimator is particularly well suited for high-frequency data analysis in financial and economic markets.<\/jats:p>","DOI":"10.1017\/s0266466619000112","type":"journal-article","created":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T07:40:35Z","timestamp":1558597235000},"page":"250-291","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":12,"title":["NONPARAMETRIC DENSITY ESTIMATION BY B-SPLINE DUALITY"],"prefix":"10.1017","volume":"36","author":[{"given":"Zhenyu","family":"Cui","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Justin Lars","family":"Kirkby","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Duy","family":"Nguyen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2019,5,23]]},"reference":[{"key":"S0266466619000112_ref25","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176349023"},{"key":"S0266466619000112_ref24","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1993.10476300"},{"key":"S0266466619000112_ref59","doi-asserted-by":"publisher","DOI":"10.1080\/10485259708832711"},{"key":"S0266466619000112_ref17","doi-asserted-by":"publisher","DOI":"10.1109\/TC.1976.1674577"},{"key":"S0266466619000112_ref74","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176345341"},{"key":"S0266466619000112_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2013.10.007"},{"key":"S0266466619000112_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejor.2017.04.007"},{"key":"S0266466619000112_ref72","doi-asserted-by":"publisher","DOI":"10.1109\/78.599940"},{"key":"S0266466619000112_ref11","first-page":"1559","article-title":"Evaluation of an unknown distribution density from observations","volume":"3","author":"Cencov","year":"1962","journal-title":"Soviet Mathematics"},{"key":"S0266466619000112_ref9","doi-asserted-by":"publisher","DOI":"10.1214\/13-AOS1158"},{"key":"S0266466619000112_ref7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2008.09.005"},{"key":"S0266466619000112_ref23","doi-asserted-by":"publisher","DOI":"10.1080\/03610919208813053"},{"key":"S0266466619000112_ref2","doi-asserted-by":"publisher","DOI":"10.2307\/2171860"},{"key":"S0266466619000112_ref65","doi-asserted-by":"publisher","DOI":"10.21314\/JCF.2015.306"},{"key":"S0266466619000112_ref51","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1018031201"},{"key":"S0266466619000112_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-8224-8"},{"key":"S0266466619000112_ref22","author":"Fix","year":"1951"},{"key":"S0266466619000112_ref8","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.2017.1285776"},{"key":"S0266466619000112_ref71","first-page":"629","volume-title":"Image Processing, 1996","volume":"1","author":"Unser","year":"1996"},{"key":"S0266466619000112_ref84","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2009.04.004"},{"key":"S0266466619000112_ref20","unstructured":"Figueroa-L\u00f3pez, J.E. & Li, C. 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