{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,3]],"date-time":"2026-09-03T14:23:54Z","timestamp":1788445434075,"version":"build-2803163510"},"reference-count":46,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2013,2,21]],"date-time":"2013-02-21T00:00:00Z","timestamp":1361404800000},"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":[[2013,10]]},"abstract":"<jats:p>Goodness-of-fit tests based on parametric empirical processes have nonstandard limiting distributions when the null hypothesis is composite \u2014 that is, when parameters of the null model are estimated. Several analytic solutions to this problem have been suggested, including the calculation of adjusted critical values for these nonstandard distributions and the transformation of the empirical process such that statistics based on the transformed process are asymptotically distribution-free. The approximation methods proposed by Durbin (1985, <jats:italic>Journal of Applied Probability<\/jats:italic> 22(1), 99\u2013122) can be applied to conduct inference for tests based on supremum-norm statistics. The resulting tests have quite accurate size, a fact that has gone unrecognized in the econometrics literature. Some justification for this accuracy lies in the similar features that Durbin\u2019s approximation methods share with the theory of extrema for Gaussian random fields and for Gauss-Markov processes. These adjustment techniques are also related to the transformation methodology proposed by Khmaladze (1981, <jats:italic>Theory of Probability and Its Applications<\/jats:italic>26(2), 240\u2013257) through the score function of the parametric model. Simulation experiments suggest that in small samples, Durbin-style adjustments result in tests that have higher power than tests based on transformed processes, and in some cases they have higher power than parametric bootstrap procedures.<\/jats:p>","DOI":"10.1017\/s0266466612000825","type":"journal-article","created":{"date-parts":[[2013,2,21]],"date-time":"2013-02-21T09:20:09Z","timestamp":1361438409000},"page":"969-1008","source":"Crossref","is-referenced-by-count":5,"title":["A COMPARISON OF ALTERNATIVE APPROACHES TO SUPREMUM-NORM GOODNESS-OF-FIT TESTS WITH ESTIMATED PARAMETERS"],"prefix":"10.1017","volume":"29","author":[{"given":"Thomas","family":"Parker","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2013,2,21]]},"reference":[{"key":"S0266466612000825_ref7","doi-asserted-by":"crossref","first-page":"2388","DOI":"10.1214\/aos\/1030741078","article-title":"Transformed empirical processes and modified Kolmogorov-Smirnov tests for multivariate distributions","volume":"25","author":"Caba\u00f1a","year":"1997","journal-title":"Annals of Statistics"},{"key":"S0266466612000825_ref11","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1017\/S0001867800010892","article-title":"A computational approach to first-passage-time problems for Gauss-Markov processes","volume":"33","author":"Di Nardo","year":"2001","journal-title":"Advances in Applied Probability"},{"key":"S0266466612000825_ref6","doi-asserted-by":"crossref","first-page":"784","DOI":"10.2307\/1427102","article-title":"A new integral equation for the evaluation of first-passage-time probability densities","volume":"19","author":"Buonocore","year":"1987","journal-title":"Advances in Applied Probability"},{"key":"S0266466612000825_ref15","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1214\/aos\/1176342365","article-title":"Weak convergence of the sample distribution function when parameters are estimated","volume":"1","author":"Durbin","year":"1973b","journal-title":"Annals of Statistics"},{"key":"S0266466612000825_ref37","first-page":"68","volume-title":"Weak Convergence Under Contiguous Alternatives When Parameters Are Estimated: The Dk Approach","volume":"566","author":"Neuhaus","year":"1976"},{"key":"S0266466612000825_ref23","volume-title":"Goodness-of-fit Testing for Duration Models with Censored Grouped Data","author":"Hong","year":"2009"},{"key":"S0266466612000825_ref2","doi-asserted-by":"crossref","first-page":"1097","DOI":"10.2307\/2171880","article-title":"A conditional Kolmogorov test","volume":"65","author":"Andrews","year":"1997","journal-title":"Econometrica"},{"key":"S0266466612000825_ref36","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1111\/j.2517-6161.1965.tb00611.x","article-title":"Certain properties of Gaussian processes and their first-passage times","volume":"27","author":"Mehr","year":"1965","journal-title":"Journal of the Royal Statistical Society, Series B (Methodological)"},{"key":"S0266466612000825_ref25","doi-asserted-by":"crossref","first-page":"995","DOI":"10.1214\/009053604000000274","article-title":"Martingale transforms goodness-of-fit tests in regression models","volume":"32","author":"Khmaladze","year":"2004","journal-title":"Annals of Statistics"},{"key":"S0266466612000825_ref40","volume-title":"Numerical Recipes in Fortran 77: The Art of Scientific Computing","author":"Press","year":"2001"},{"key":"S0266466612000825_ref45","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1214\/074921707000000382","volume-title":"Asymptotics: Particles, Processes and Inverse Problems","volume":"55","author":"van der Vaart","year":"2007"},{"key":"S0266466612000825_ref3","doi-asserted-by":"crossref","DOI":"10.1002\/9780470434642","volume-title":"Level Sets and Extrema of Random Processes and Fields","author":"Aza\u00efs","year":"2009"},{"key":"S0266466612000825_ref42","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.jeconom.2009.07.002","article-title":"Testing semiparametric conditional moment restrictions using conditional martingale transforms","volume":"154","author":"Song","year":"2010","journal-title":"Journal of Econometrics"},{"key":"S0266466612000825_ref22","volume-title":"Generalized Residual-based Specification Testing for Duration Models with Censoring","author":"Hong","year":"2007"},{"key":"S0266466612000825_ref32","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1080\/07474930903038933","article-title":"Asymptotically distribution-free goodness-of-fit testing: A unifying view","volume":"28","author":"Li","year":"2009","journal-title":"Econometric Reviews"},{"key":"S0266466612000825_ref13","doi-asserted-by":"crossref","first-page":"431","DOI":"10.2307\/3212169","article-title":"Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test","volume":"8","author":"Durbin","year":"1971","journal-title":"Journal of Applied Probability"},{"key":"S0266466612000825_ref26","doi-asserted-by":"crossref","first-page":"3165","DOI":"10.1214\/08-AOS680","article-title":"Goodness-of-fit problem for errors in nonparametric regression: Distribution free approach","volume":"37","author":"Khmaladze","year":"2009","journal-title":"Annals of Statistics"},{"key":"S0266466612000825_ref14","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970586","volume-title":"Distribution Theory for Tests Based on the Sample Distribution Function","author":"Durbin","year":"1973a"},{"key":"S0266466612000825_ref34","unstructured":"Martynov, G. (2009) Goodness-of-fit Tests for the Weibull and Pareto Distributions. Paper presented at the Sixth International Conference on Mathematical Methods in Reliability."},{"key":"S0266466612000825_ref4","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1162\/003465303322369704","article-title":"Testing parametric conditional distributions of dynamic models","volume":"85","author":"Bai","year":"2003","journal-title":"Review of Economics and Statistics"},{"key":"S0266466612000825_ref35","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/BF02506887","article-title":"Empirical characteristic function approach to goodness-of-fit tests for the Cauchy distribution with parameters estimated by MLE or EISE","volume":"57","author":"Matsui","year":"2005","journal-title":"Annals of the Institute of Statistical Mathematics"},{"key":"S0266466612000825_ref33","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1214\/aos\/1176344954","article-title":"The empirical distribution function of residuals from generalised regression","volume":"8","author":"Loynes","year":"1980","journal-title":"Annals of Statistics"},{"key":"S0266466612000825_ref30","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1016\/j.spl.2005.04.053","article-title":"Goodness-of-fit testing in regression: A finite sample comparison of bootstrap methodology and Khmaladze transformation","volume":"74","author":"Koul","year":"2005","journal-title":"Statistics and Probability Letters"},{"key":"S0266466612000825_ref5","volume-title":"Efficient and Adaptive Estimation for Semiparametric Models","author":"Bickel","year":"1993"},{"key":"S0266466612000825_ref17","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1017\/S0021900200029041","article-title":"The first-passage density of a continuous Gaussian process to a general boundary","volume":"22","author":"Durbin","year":"1985","journal-title":"Journal of Applied Probability"},{"key":"S0266466612000825_ref43","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/BF02613687","article-title":"Bootstrap based goodness-of-fit-tests","volume":"40","author":"Stute","year":"1993","journal-title":"Metrika"},{"key":"S0266466612000825_ref16","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1093\/biomet\/62.1.5","article-title":"Kolmogorov-Smirnov tests when parameters are estimated with applications to tests of exponentiality and tests on spacings","volume":"62","author":"Durbin","year":"1975","journal-title":"Biometrika"},{"key":"S0266466612000825_ref20","first-page":"21","article-title":"Asymptotics of large deviation probabilities for Gaussian fields: Applications","volume":"28","author":"Fatalov","year":"1993","journal-title":"Journal of Contemporary Mathematical Analysis"},{"key":"S0266466612000825_ref44","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2545-2","volume-title":"Weak Convergence and Empirical Processes","author":"van der Vaart","year":"1996"},{"key":"S0266466612000825_ref21","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1016\/j.jeconom.2007.08.005","article-title":"On distribution-free goodness-of-fit testing of exponentiality","volume":"143","author":"Haywood","year":"2008","journal-title":"Journal of Econometrics"},{"key":"S0266466612000825_ref27","doi-asserted-by":"crossref","first-page":"1583","DOI":"10.1111\/1468-0262.00342","article-title":"Inference on the quantile regression process","volume":"70","author":"Koenker","year":"2002","journal-title":"Econometrica"},{"key":"S0266466612000825_ref12","volume-title":"Stochastic Processes","author":"Doob","year":"1953"},{"key":"S0266466612000825_ref8","first-page":"183","article-title":"Heteroskedasticity-robust tests in regression directions","volume":"59","author":"Davidson","year":"1985","journal-title":"Annales de l\u2019INSEE"},{"key":"S0266466612000825_ref9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4171\/027","volume-title":"Lectures on Empirical Processes: Theory and Statistical Applications","author":"del Barrio","year":"2007"},{"key":"S0266466612000825_ref19","first-page":"48","article-title":"Asymptotics of large deviation probabilities for Gaussian fields","volume":"27","author":"Fatalov","year":"1992","journal-title":"Journal of Contemporary Mathematical Analysis"},{"key":"S0266466612000825_ref31","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1080\/10485259508832633","article-title":"Coefficients of the asymptotic distribution of the Kolmogorov-Smirnov statistic when parameters are estimated","volume":"5","author":"Kulinskaya","year":"1995","journal-title":"Journal of Nonparametric Statistics"},{"key":"S0266466612000825_ref39","volume-title":"Asymptotic Methods in the Theory of Gaussian Processes and Fields","volume":"148","author":"Piterbarg","year":"1996"},{"key":"S0266466612000825_ref28","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4613-0055-7","volume-title":"Weighted Empirical Processes in Dynamic Nonlinear Models","volume":"166","author":"Koul","year":"2002"},{"key":"S0266466612000825_ref1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02482496","article-title":"Some test statistics based on the martingale term of the empirical distribution function","volume":"38","author":"Aki","year":"1986","journal-title":"Annals of the Institute of Statistical Mathematics"},{"key":"S0266466612000825_ref29","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1142\/9781860948886_0009","volume-title":"Frontiers in Statistics","author":"Koul","year":"2006"},{"key":"S0266466612000825_ref41","volume-title":"Empirical Processes with Applications to Statistics","author":"Shorack","year":"1986"},{"key":"S0266466612000825_ref10","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.jeconom.2007.08.006","article-title":"Distribution-free specification tests of conditional models","volume":"143","author":"Delgado","year":"2008","journal-title":"Journal of Econometrics"},{"key":"S0266466612000825_ref38","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1216\/jiea\/1181074930","article-title":"On integral equations arising in the first-passage problem for Brownian motion","volume":"14","author":"Peskir","year":"2002","journal-title":"Journal of Integral Equations and Applications"},{"key":"S0266466612000825_ref18","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1111\/j.2517-6161.1975.tb01537.x","article-title":"Components of the Cram\u00e9r-von Mises statistics II.","volume":"37","author":"Durbin","year":"1975","journal-title":"Journal of the Royal Statistical Society, Series B (Methodological)"},{"key":"S0266466612000825_ref46","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1017\/S0266466600004898","article-title":"A unified approach to robust, regression-based specification tests","volume":"6","author":"Wooldridge","year":"1990","journal-title":"Econometric Theory"},{"key":"S0266466612000825_ref24","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1137\/1126027","article-title":"Martingale approach in the theory of goodness-of-fit tests","volume":"26","author":"Khmaladze","year":"1981","journal-title":"Theory of Probability and Its Applications"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466612000825","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,12,11]],"date-time":"2020-12-11T23:02:58Z","timestamp":1607727778000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466612000825\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,2,21]]},"references-count":46,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2013,10]]}},"alternative-id":["S0266466612000825"],"URL":"https:\/\/doi.org\/10.1017\/s0266466612000825","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,2,21]]}}}