{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,2]],"date-time":"2025-12-02T03:27:16Z","timestamp":1764646036034},"reference-count":55,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2014,4,11]],"date-time":"2014-04-11T00:00:00Z","timestamp":1397174400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2014,10]]},"abstract":"<jats:p>This paper considers sieve maximum likelihood estimation of seminonparametric (SNP) models with an unknown density function as non-Euclidean parameter, next to a finite-dimensional parameter vector. The density function involved is modeled via an infinite series expansion, so that the actual parameter space is infinite-dimensional. It will be shown that under low-level conditions the sieve estimators of these parameters are consistent, and the estimators of the Euclidean parameters are<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0266466614000036_inline1\"\/><jats:tex-math>$\\sqrt N$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>asymptotically normal, given a random sample of size<jats:italic>N<\/jats:italic>. The latter result is derived in a different way than in the sieve estimation literature. It appears that this asymptotic normality result is in essence the same as for the finite dimensional case. This approach is motivated and illustrated by an SNP discrete choice model.<\/jats:p>","DOI":"10.1017\/s0266466614000036","type":"journal-article","created":{"date-parts":[[2014,4,11]],"date-time":"2014-04-11T12:52:00Z","timestamp":1397220720000},"page":"1021-1076","source":"Crossref","is-referenced-by-count":12,"title":["CONSISTENCY AND ASYMPTOTIC NORMALITY OF SIEVE ML ESTIMATORS UNDER LOW-LEVEL CONDITIONS"],"prefix":"10.1017","volume":"30","author":[{"given":"Herman J.","family":"Bierens","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,4,11]]},"reference":[{"key":"S0266466614000036_ref5","volume-title":"Efficient and Adaptive Estimation for Semiparametric Models","author":"Bickel","year":"1998"},{"key":"S0266466614000036_ref31","doi-asserted-by":"publisher","DOI":"10.2307\/1911491"},{"key":"S0266466614000036_ref22","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1080\/07350015.1993.10509933","article-title":"Seminonparametric estimation of binary-choice models with an application to labor-force participation","volume":"11","author":"Gabler","year":"1993","journal-title":"Journal of Business and Economic Statistics"},{"key":"S0266466614000036_ref27","volume-title":"Abstract Inference","author":"Grenander","year":"1981"},{"key":"S0266466614000036_ref19","doi-asserted-by":"publisher","DOI":"10.2307\/1912157"},{"key":"S0266466614000036_ref18","doi-asserted-by":"publisher","DOI":"10.1111\/1468-0262.00461"},{"key":"S0266466614000036_ref15","doi-asserted-by":"crossref","unstructured":"Chen X . (2011) Penalized Sieve Estimation and Inference of Semi-nonparametric Dynamic Models: A Selective Review. Cowles Foundation Discussion paper no. 2804, Yale University.","DOI":"10.1920\/wp.cem.2011.2311"},{"key":"S0266466614000036_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2007.01.013"},{"key":"S0266466614000036_ref43","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1176996608"},{"key":"S0266466614000036_ref25","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176345782"},{"key":"S0266466614000036_ref49","doi-asserted-by":"crossref","first-page":"2555","DOI":"10.1214\/aos\/1030741085","article-title":"On the method of sieves and penalization","volume":"25","author":"Shen","year":"1997","journal-title":"Annals of Statistics"},{"key":"S0266466614000036_ref13","volume-title":"Convergence of Probability Measures","author":"Billingsley","year":"1968"},{"key":"S0266466614000036_ref35","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2010.05.005"},{"key":"S0266466614000036_ref8","doi-asserted-by":"crossref","unstructured":"Bierens H.J . (2013) Supplement to: Consistency and asymptotic normality of sieve ML estimators under low-level conditions. http:\/\/econ.la.psu.edu\/\u223chbierens\/SNPMODELS_A.PDF","DOI":"10.1017\/S0266466614000164"},{"key":"S0266466614000036_ref39","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1080\/01621459.1968.11009321","article-title":"The estimation of densities and cumulatives by Fourier series methods","volume":"63","author":"Kronmal","year":"1968","journal-title":"Journal of the American Statistical Association"},{"key":"S0266466614000036_ref29","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1111\/j.2517-6161.1979.tb01072.x","article-title":"The determination of the order of an autoregression","volume":"41","author":"Hannan","year":"1979","journal-title":"Journal of the Royal Statistical Society, Series B"},{"key":"S0266466614000036_ref36","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177697731"},{"key":"S0266466614000036_ref44","doi-asserted-by":"publisher","DOI":"10.1002\/jae.3950050202"},{"key":"S0266466614000036_ref30","doi-asserted-by":"publisher","DOI":"10.2307\/1912352"},{"key":"S0266466614000036_ref42","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1988.10478655"},{"key":"S0266466614000036_ref20","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600004497"},{"key":"S0266466614000036_ref45","doi-asserted-by":"publisher","DOI":"10.2307\/2951752"},{"key":"S0266466614000036_ref52","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511802256"},{"key":"S0266466614000036_ref23","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4076(81)90115-9"},{"key":"S0266466614000036_ref16","doi-asserted-by":"publisher","DOI":"10.1198\/016214506000000311"},{"key":"S0266466614000036_ref9","first-page":"3","volume-title":"Oxford Handbook of Applied Nonparametric and Semiparametric Econometrics and Statistics","author":"Bierens","year":"2014"},{"key":"S0266466614000036_ref47","doi-asserted-by":"publisher","DOI":"10.1111\/1468-0262.00460"},{"key":"S0266466614000036_ref37","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2012.08.002"},{"key":"S0266466614000036_ref40","doi-asserted-by":"publisher","DOI":"10.2307\/1914140"},{"key":"S0266466614000036_ref24","doi-asserted-by":"publisher","DOI":"10.2307\/1913241"},{"key":"S0266466614000036_ref21","doi-asserted-by":"publisher","DOI":"10.2307\/2297364"},{"key":"S0266466614000036_ref1","doi-asserted-by":"publisher","DOI":"10.1162\/REST_a_00251"},{"key":"S0266466614000036_ref46","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-4076(97)00011-0"},{"key":"S0266466614000036_ref34","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4076(93)90114-K"},{"key":"S0266466614000036_ref32","doi-asserted-by":"publisher","DOI":"10.2307\/2951582"},{"key":"S0266466614000036_ref38","doi-asserted-by":"publisher","DOI":"10.2307\/2951556"},{"key":"S0266466614000036_ref14","doi-asserted-by":"crossref","first-page":"5549","DOI":"10.1016\/S1573-4412(07)06076-X","volume-title":"Handbook of Econometrics","author":"Chen","year":"2007"},{"key":"S0266466614000036_ref50","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177693161"},{"key":"S0266466614000036_ref11","unstructured":"Bierens H.J. & Song H. (2013) Semi-nonparametric Modeling and Estimation of First-Price Auctions Models with Auction-Specific Heterogeneity. Working paper, Penn State University. http:\/\/econ.la.psu.edu\/\u223chbierens\/AUCTIONS_HETERO.PDF"},{"key":"S0266466614000036_ref10","doi-asserted-by":"publisher","DOI":"10.1002\/jae.960"},{"key":"S0266466614000036_ref7","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466608080316"},{"key":"S0266466614000036_ref17","doi-asserted-by":"publisher","DOI":"10.2307\/2998559"},{"key":"S0266466614000036_ref54","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177729952"},{"key":"S0266466614000036_ref3","doi-asserted-by":"publisher","DOI":"10.2307\/2951475"},{"key":"S0266466614000036_ref51","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1177\/1536867X0100400102","article-title":"Semi-nonparametric estimation of extended ordered Probit models","volume":"4","author":"Stewart","year":"2004","journal-title":"The Stata Journal"},{"key":"S0266466614000036_ref6","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511754012"},{"key":"S0266466614000036_ref41","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4076(85)90009-0"},{"key":"S0266466614000036_ref53","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2545-2"},{"key":"S0266466614000036_ref28","doi-asserted-by":"publisher","DOI":"10.2307\/2297911"},{"key":"S0266466614000036_ref33","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0621-7"},{"key":"S0266466614000036_ref4","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177698949"},{"key":"S0266466614000036_ref48","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176344136"},{"key":"S0266466614000036_ref55","first-page":"459","volume-title":"Non-parametric and Semi-parametric Methods in Econometrics and Statistics","author":"White","year":"1991"},{"key":"S0266466614000036_ref12","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466611000168"},{"key":"S0266466614000036_ref26","first-page":"97","article-title":"Non- and semi-parametric maximum likelihood estimators and the von Mises method (Part 1)","volume":"16","author":"Gill","year":"1989","journal-title":"Scandinavian Journal of Statistics"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466614000036","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,25]],"date-time":"2024-05-25T21:04:00Z","timestamp":1716671040000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466614000036\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,4,11]]},"references-count":55,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2014,10]]}},"alternative-id":["S0266466614000036"],"URL":"https:\/\/doi.org\/10.1017\/s0266466614000036","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,4,11]]}}}