{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T05:39:02Z","timestamp":1791524342978,"version":"4.3.4"},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2009,2,11]],"date-time":"2009-02-11T00:00:00Z","timestamp":1234310400000},"content-version":"unspecified","delay-in-days":5885,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[1993,1]]},"abstract":"<jats:p>The smoothed maximum score estimator of the coefficient vector of a binary response model is consistent and, after centering and suitable normalization, asymptotically normally distributed under weak assumptions [5]. Its rate of convergence in probability is <jats:italic>N<\/jats:italic><jats:sup>\u2212<jats:italic>h<\/jats:italic>\/(2<jats:italic>h<\/jats:italic>+1)<\/jats:sup>, where <jats:italic>h<\/jats:italic> \u2265 2 is an integer whose value depends on the strength of certain smoothness assumptions. This rate of convergence is faster than that of the maximum score estimator of Manski [11,12], which converges at the rate <jats:italic>N<\/jats:italic><jats:sup>\u22121\/3<\/jats:sup> under assumptions that are somewhat weaker than those of the smoothed estimator. In this paper I prove that under the assumptions of smoothed maximum score estimation, <jats:italic>N<\/jats:italic><jats:sup>\u2212<jats:italic>h<\/jats:italic>\/(2<jats:italic>h<\/jats:italic>+1)<\/jats:sup> is the fastest achievable rate of convergence of an estimator of the coefficient vector of a binary response model. Thus, the smoothed maximum score estimator has the fastest possible rate of convergence. The rate of convergence is defined in a minimax sense so as to exclude superefficient estimators.<\/jats:p>","DOI":"10.1017\/s0266466600007301","type":"journal-article","created":{"date-parts":[[2009,3,10]],"date-time":"2009-03-10T11:42:54Z","timestamp":1236685374000},"page":"1-18","source":"Crossref","is-referenced-by-count":20,"title":["Optimal Rates of Convergence of Parameter Estimators in the Binary Response Model with Weak Distributional Assumptions"],"prefix":"10.1017","volume":"9","author":[{"given":"Joel L.","family":"Horowitz","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2009,2,11]]},"reference":[{"key":"S0266466600007301_ref003","first-page":"175","volume-title":"Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability","author":"Hajek","year":"1972"},{"key":"S0266466600007301_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4076(86)90038-2"},{"key":"S0266466600007301_ref012","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4076(85)90009-0"},{"key":"S0266466600007301_ref016","doi-asserted-by":"publisher","DOI":"10.2307\/1914309"},{"key":"S0266466600007301_ref015","doi-asserted-by":"publisher","DOI":"10.2307\/1913713"},{"key":"S0266466600007301_ref007","volume-title":"Semiparametric least squares estimation of single index models","author":"Ichimura","year":"1988"},{"key":"S0266466600007301_ref010","first-page":"277","article-title":"On some asymptotic properties of maximum likelihood estimates and related Bayes' estimates","volume":"1","author":"LeCam","year":"1953","journal-title":"University of California Publications in Statistics"},{"key":"S0266466600007301_ref014","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176346523"},{"key":"S0266466600007301_ref009","volume-title":"An efficient semiparametric estimator for discrete choice models","author":"Klein","year":"1989"},{"key":"S0266466600007301_ref008","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176347498"},{"key":"S0266466600007301_ref005","doi-asserted-by":"publisher","DOI":"10.2307\/2951582"},{"key":"S0266466600007301_ref002","doi-asserted-by":"publisher","DOI":"10.2307\/1912157"},{"key":"S0266466600007301_ref017","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176345206"},{"key":"S0266466600007301_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-0027-2"},{"key":"S0266466600007301_ref013","first-page":"237","volume-title":"Frontiers in Econometrics","author":"McFadden","year":"1974"},{"key":"S0266466600007301_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4076(75)90032-9"},{"key":"S0266466600007301_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4076(87)90030-3"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466600007301","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T15:28:32Z","timestamp":1558020512000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466600007301\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,1]]},"references-count":17,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1993,1]]}},"alternative-id":["S0266466600007301"],"URL":"https:\/\/doi.org\/10.1017\/s0266466600007301","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,1]]}}}