{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T22:10:14Z","timestamp":1780697414789,"version":"3.54.1"},"reference-count":49,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2017,12,1]],"date-time":"2017-12-01T00:00:00Z","timestamp":1512086400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.cambridge.org"],"crossmark-restriction":true},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2018,10]]},"abstract":"<jats:p>In this article, we propose a general method for testing inequality restrictions on nonparametric functions. Our framework includes many nonparametric testing problems in a unified framework, with a number of possible applications in auction models, game theoretic models, wage inequality, and revealed preferences. Our test involves a one-sided version of<jats:italic>L<\/jats:italic><jats:sub><jats:italic>p<\/jats:italic><\/jats:sub>functionals of kernel-type estimators (1 \u2264<jats:italic>p &lt; \u221e<\/jats:italic>) and is easy to implement in general, mainly due to its recourse to the bootstrap method. The bootstrap procedure is based on the nonparametric bootstrap applied to kernel-based test statistics, with an option of estimating \u201ccontact sets.\u201d We provide regularity conditions under which the bootstrap test is asymptotically valid uniformly over a large class of distributions, including cases where the limiting distribution of the test statistic is degenerate. Our bootstrap test is shown to exhibit good power properties in Monte Carlo experiments, and we provide a general form of the local power function. As an illustration, we consider testing implications from auction theory, provide primitive conditions for our test, and demonstrate its usefulness by applying our test to real data. We supplement this example with the second empirical illustration in the context of wage inequality.<\/jats:p>","DOI":"10.1017\/s0266466617000329","type":"journal-article","created":{"date-parts":[[2017,12,1]],"date-time":"2017-12-01T09:02:08Z","timestamp":1512118928000},"page":"1018-1064","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":25,"title":["TESTING FOR A GENERAL CLASS OF FUNCTIONAL INEQUALITIES"],"prefix":"10.1017","volume":"34","author":[{"given":"Sokbae","family":"Lee","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kyungchul","family":"Song","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yoon-Jae","family":"Whang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2017,12,1]]},"reference":[{"key":"S0266466617000329_ref9","doi-asserted-by":"crossref","unstructured":"Armstrong T.B . (2014a) On the choice of test statistic for conditional moment inequalities. ArXiv:1410.4718.","DOI":"10.2139\/ssrn.2512394"},{"key":"S0266466617000329_ref21","article-title":"Adaptive tests of conditional moment inequalities","author":"Chetverikov","year":"2017","journal-title":"Econometric Theory"},{"key":"S0266466617000329_ref47","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511530081"},{"key":"S0266466617000329_ref41","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2009.08.002"},{"key":"S0266466617000329_ref15","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v14-722"},{"key":"S0266466617000329_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2014.04.021"},{"key":"S0266466617000329_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2012.05.001"},{"key":"S0266466617000329_ref24","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/996986504"},{"key":"S0266466617000329_ref16","doi-asserted-by":"publisher","DOI":"10.1214\/009053606000000137"},{"key":"S0266466617000329_ref19","doi-asserted-by":"crossref","first-page":"667","DOI":"10.3982\/ECTA8718","article-title":"Intersection bounds: Estimation and inference","volume":"81","author":"Chernozhukov","year":"2013","journal-title":"Econometrica"},{"key":"S0266466617000329_ref3","doi-asserted-by":"crossref","first-page":"2247","DOI":"10.1016\/S1573-4412(05)80006-6","volume-title":"The Handbook of Econometrics","volume":"vol. 4","author":"Andrews","year":"1994"},{"key":"S0266466617000329_ref31","doi-asserted-by":"publisher","DOI":"10.1086\/344801"},{"key":"S0266466617000329_ref32","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176349403"},{"key":"S0266466617000329_ref22","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2012.02.005"},{"key":"S0266466617000329_ref18","doi-asserted-by":"publisher","DOI":"10.1111\/1468-0262.00461"},{"key":"S0266466617000329_ref46","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2014.05.016"},{"key":"S0266466617000329_ref30","doi-asserted-by":"crossref","unstructured":"Haile P. , Hong H. , & Shum M. (2003) Nonparametric Tests for Common Values at First- Price Sealed-Bid Auctions. Working paper, Yale University.","DOI":"10.3386\/w10105"},{"key":"S0266466617000329_ref42","doi-asserted-by":"publisher","DOI":"10.1002\/jae.1032"},{"key":"S0266466617000329_ref40","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2012.08.006"},{"key":"S0266466617000329_ref25","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2013.08.005"},{"key":"S0266466617000329_ref45","doi-asserted-by":"publisher","DOI":"10.1214\/08-AAP569"},{"key":"S0266466617000329_ref36","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2010.04.006"},{"key":"S0266466617000329_ref48","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1080\/00031305.1998.10480576","article-title":"Expect the unexpected from conditional expectation","volume":"52","author":"Proschan","year":"1998","journal-title":"American Statistician"},{"key":"S0266466617000329_ref17","doi-asserted-by":"publisher","DOI":"10.1111\/ectj.12050"},{"key":"S0266466617000329_ref43","unstructured":"Mammen E. , Van Keilegom I. , & Yu K. (2013) Expansion for moments of regression quantiles with application to nonparametric testing. Working paper, arXiv:1306.6179."},{"key":"S0266466617000329_ref35","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1021379863"},{"key":"S0266466617000329_ref7","doi-asserted-by":"crossref","first-page":"489","DOI":"10.3982\/ECTA9431","article-title":"Identification and inference in ascending auctions with correlated private values","volume":"81","author":"Aradillas-L\u00f3pez","year":"2013","journal-title":"Econometrica"},{"key":"S0266466617000329_ref20","doi-asserted-by":"crossref","unstructured":"Chetverikov D . (2012) Testing Regression Monotonicity in Econometric Models. Working paper, MIT.","DOI":"10.1920\/wp.cem.2012.3512"},{"key":"S0266466617000329_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2016.04.006"},{"key":"S0266466617000329_ref23","doi-asserted-by":"publisher","DOI":"10.1080\/07350015.2012.723556"},{"key":"S0266466617000329_ref38","unstructured":"Krasnokutskaya E. , Song K. , & Tang X. (2016) The Role of Quality in On-Line Service Markets. Working paper, Johns Hopkins University."},{"key":"S0266466617000329_ref37","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2009.01.006"},{"key":"S0266466617000329_ref27","unstructured":"Gimenes N. & Guerre E. (2013) Augmented Quantile Regression Methods for First-Price Auction. Working paper, Queen Mary University of London."},{"key":"S0266466617000329_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2016.04.001"},{"key":"S0266466617000329_ref6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2016.09.010"},{"key":"S0266466617000329_ref26","doi-asserted-by":"crossref","first-page":"1054","DOI":"10.1214\/aos\/1015956707","article-title":"Testing monotonicity of regression","volume":"28","author":"Ghosal","year":"2000","journal-title":"Annals of Statistics"},{"key":"S0266466617000329_ref44","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2013.04.005"},{"key":"S0266466617000329_ref49","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176342996"},{"key":"S0266466617000329_ref5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2013.10.005"},{"key":"S0266466617000329_ref1","first-page":"1043","volume-title":"Handbook of Labor Economics","volume":"vol. 4","author":"Acemoglu","year":"2011"},{"key":"S0266466617000329_ref34","doi-asserted-by":"publisher","DOI":"10.1111\/ectj.12077"},{"key":"S0266466617000329_ref14","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466614000401"},{"key":"S0266466617000329_ref4","doi-asserted-by":"crossref","first-page":"609","DOI":"10.3982\/ECTA9370","article-title":"Inference based on conditional moment inequalities","volume":"81","author":"Andrews","year":"2013","journal-title":"Econometrica"},{"key":"S0266466617000329_ref28","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1214\/aop\/1048516534","article-title":"The L1-norm density estimator process","volume":"31","author":"Gin\u00e9","year":"2003","journal-title":"Annals of Probability"},{"key":"S0266466617000329_ref33","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176348379"},{"key":"S0266466617000329_ref29","doi-asserted-by":"crossref","first-page":"1193","DOI":"10.3982\/ECTA7028","article-title":"Nonparametric identification of risk aversion in first-price auctions under exclusion restrictions","volume":"77","author":"Guerre","year":"2009","journal-title":"Econometrica"},{"key":"S0266466617000329_ref39","doi-asserted-by":"crossref","first-page":"585","DOI":"10.3982\/ECTA7145","article-title":"Testing for stochastic monotonicity","volume":"77","author":"Lee","year":"2009","journal-title":"Econometrica"},{"key":"S0266466617000329_ref13","doi-asserted-by":"publisher","DOI":"10.1214\/009053604000000896"},{"key":"S0266466617000329_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2015.01.002"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466617000329","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,29]],"date-time":"2024-06-29T08:59:45Z","timestamp":1719651585000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466617000329\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12,1]]},"references-count":49,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2018,10]]}},"alternative-id":["S0266466617000329"],"URL":"https:\/\/doi.org\/10.1017\/s0266466617000329","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,12,1]]},"assertion":[{"value":"Copyright \u00a9 Cambridge University Press 2017\u00a0This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}