{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,21]],"date-time":"2025-06-21T00:40:10Z","timestamp":1750466410074,"version":"3.41.0"},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2003,9,24]],"date-time":"2003-09-24T00:00:00Z","timestamp":1064361600000},"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":[[2003,12]]},"abstract":"<jats:p>Let \n<jats:inline-formula>\n\t      <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:href=\"S0266466603196090ffm053.gif\"\/>\n\t    <\/jats:inline-formula> \nand \u0398 be infinite sets and let \n<jats:inline-formula>\n\t      <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:href=\"S0266466603196090ffm054.gif\"\/>\n\t    <\/jats:inline-formula> \n\u00d7 \u0398. We show that the class of projections of <jats:italic>A<\/jats:italic> onto \n<jats:inline-formula>\n\t      <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:href=\"S0266466603196090ffm055.gif\"\/>\n\t    <\/jats:inline-formula> \nis a Vapnik\u2013Chervonenkis (<jats:italic>VC<\/jats:italic>) class of sets if and only if \nthe class of projections of <jats:italic>A<\/jats:italic> onto \u0398 is a <jats:italic>VC<\/jats:italic> class. \nWe illustrate the result in the context of semiparametric estimation of a \ntransformation model. In this application, the <jats:italic>VC<\/jats:italic> property is hard \nto establish for the projection class of interest but easy to establish for \nthe other projection class.<\/jats:p>","DOI":"10.1017\/s0266466603196090","type":"journal-article","created":{"date-parts":[[2003,11,3]],"date-time":"2003-11-03T15:52:13Z","timestamp":1067874733000},"page":"1123-1127","source":"Crossref","is-referenced-by-count":2,"title":["AN EQUIVALENCE RESULT FOR <i>VC<\/i> CLASSES OF SETS"],"prefix":"10.1017","volume":"19","author":[{"given":"Scott","family":"Joslin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert P.","family":"Sherman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2003,9,24]]},"reference":[{"key":"S0266466603196090_ref013","doi-asserted-by":"crossref","unstructured":"Pakes, A. & D. Pollard (1989)Simulation and the asymptotics of optimization estimators.Econometrica 57,1027\u20131057.","DOI":"10.2307\/1913622"},{"key":"S0266466603196090_ref011","doi-asserted-by":"crossref","unstructured":"Manski, C. (1975)Maximum score estimation of the stochastic utility model of choice.Journal of Econometrics 3,205\u2013228.","DOI":"10.1016\/0304-4076(75)90032-9"},{"key":"S0266466603196090_ref016","doi-asserted-by":"crossref","unstructured":"Stengle, G. & J.E. Yukich (1989)Some new Vapnik\u2013Chervonenkis classes.Annals of Statistics 17,1441\u20131446.","DOI":"10.1214\/aos\/1176347373"},{"key":"S0266466603196090_ref007","doi-asserted-by":"crossref","unstructured":"Chen, S. (2002)Rank estimation of transformation models.Econometrica 70,1683\u20131697.","DOI":"10.1111\/1468-0262.00347"},{"key":"S0266466603196090_ref006","doi-asserted-by":"crossref","unstructured":"Cavanagh, C.L. & R.P. Sherman (1998)Rank estimators for monotonic index models.Journal of Econometrics 84,351\u2013381.","DOI":"10.1016\/S0304-4076(97)00090-0"},{"key":"S0266466603196090_ref014","doi-asserted-by":"crossref","unstructured":"Pollard, D. (1984)Convergence of Stochastic Processes.New York:Springer-Verlag.","DOI":"10.1007\/978-1-4612-5254-2"},{"key":"S0266466603196090_ref010","doi-asserted-by":"crossref","unstructured":"Khan, S. (2001)Two stage rank estimation of quantile index models.Journal of Econometrics 100,319\u2013355.","DOI":"10.1016\/S0304-4076(00)00040-3"},{"key":"S0266466603196090_ref018","doi-asserted-by":"crossref","unstructured":"Wenocur, R.S. & R.M. Dudley (1981)Some special Vapnik\u2013Chervonenkis classes.Discrete Mathematics 33,313\u2013318.","DOI":"10.1016\/0012-365X(81)90274-0"},{"key":"S0266466603196090_ref001","doi-asserted-by":"crossref","unstructured":"Abrevaya, J. (1999)Leapfrog estimation of a fixed-effects model with unknowntransformation of the dependent variable.Journal of Econometrics 93,203\u2013228.","DOI":"10.1016\/S0304-4076(99)00009-3"},{"key":"S0266466603196090_ref005","doi-asserted-by":"crossref","unstructured":"Box, G.E.P. & D.R. Cox (1964)An analysis of transformations.Journal of the Royal Statistical Society, Series B 34,187\u2013220.","DOI":"10.1111\/j.2517-6161.1964.tb00553.x"},{"key":"S0266466603196090_ref009","doi-asserted-by":"crossref","unstructured":"Han, A.K. (1987b)Non-parametric analysis of a generalized regression model.Journal of Econometrics 35,303\u2013316.","DOI":"10.1016\/0304-4076(87)90030-3"},{"key":"S0266466603196090_ref003","doi-asserted-by":"crossref","unstructured":"Andrews, D.W. (1994)Empirical process methods in econometrics. In R.F. Engle & D.L. McFadden (eds.),Handbook of Econometrics,vol. 4, pp.2247\u20132294.Amsterdam:North-Holland.","DOI":"10.1016\/S1573-4412(05)80006-6"},{"key":"S0266466603196090_ref004","doi-asserted-by":"crossref","unstructured":"Asparouhova, E. , T. Asparouhov , R. Golanski , K. Kasprzyk , & R.P. Sherman (2002)Rank estimators for a transformation model.Econometric Theory 18,1099\u20131120.","DOI":"10.1017\/S0266466602185045"},{"key":"S0266466603196090_ref008","doi-asserted-by":"crossref","unstructured":"Han, A.K. (1987a)A nonparametric analysis of transformations.Journal of Econometrics 35,191\u2013209.","DOI":"10.1016\/0304-4076(87)90023-6"},{"key":"S0266466603196090_ref015","doi-asserted-by":"crossref","unstructured":"Pollard, D. (1989)Asymptotics via empirical processes.Statistical Science 4,341\u2013366.","DOI":"10.1214\/ss\/1177012394"},{"key":"S0266466603196090_ref017","doi-asserted-by":"crossref","unstructured":"van den Dries, L. (1998)Tame Topology and O-minimal Structures.Cambridge:Cambridge University Press.","DOI":"10.1017\/CBO9780511525919"},{"key":"S0266466603196090_ref012","doi-asserted-by":"crossref","unstructured":"Manski, C. (1985)Semiparametric analysis of discrete response: Asymptotic propertiesof the maximum score estimator.Journal of Econometrics 27,313\u2013334.","DOI":"10.1016\/0304-4076(85)90009-0"},{"key":"S0266466603196090_ref002","unstructured":"Abrevaya, J. (2002)Pairwise-Difference Rank Estimation of the Transformation Model.Manuscript,University of Chicago Graduate School of Business."}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466603196090","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,21]],"date-time":"2025-06-21T00:04:31Z","timestamp":1750464271000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466603196090\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,9,24]]},"references-count":18,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2003,12]]}},"alternative-id":["S0266466603196090"],"URL":"https:\/\/doi.org\/10.1017\/s0266466603196090","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"type":"print","value":"0266-4666"},{"type":"electronic","value":"1469-4360"}],"subject":[],"published":{"date-parts":[[2003,9,24]]}}}