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Continuity of the Distribution Function of the argmax of a Gaussian Process

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Listed:
  • Matias D. Cattaneo
  • Gregory Fletcher Cox
  • Michael Jansson
  • Kenichi Nagasawa

Abstract

An increasingly important class of estimators has members whose asymptotic distribution is non-Gaussian, yet characterizable as the argmax of a Gaussian process. This paper presents high-level sufficient conditions under which such asymptotic distributions admit a continuous distribution function. The plausibility of the sufficient conditions is demonstrated by verifying them in three prominent examples, namely maximum score estimation, empirical risk minimization, and threshold regression estimation. In turn, the continuity result buttresses several recently proposed inference procedures whose validity seems to require a result of the kind established herein. A notable feature of the high-level assumptions is that one of them is designed to enable us to employ the celebrated Cameron-Martin theorem. In a leading special case, the assumption in question is demonstrably weak and appears to be close to minimal.

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  • Matias D. Cattaneo & Gregory Fletcher Cox & Michael Jansson & Kenichi Nagasawa, 2025. "Continuity of the Distribution Function of the argmax of a Gaussian Process," Papers 2501.13265, arXiv.org.
  • Handle: RePEc:arx:papers:2501.13265
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    References listed on IDEAS

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    1. Lai, P.Y. & Lee, Stephen M.S., 2005. "An Overview of Asymptotic Properties of Lp Regression Under General Classes of Error Distributions," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 446-458, June.
    2. Delgado, Miguel A. & Rodriguez-Poo, Juan M. & Wolf, Michael, 2001. "Subsampling inference in cube root asymptotics with an application to Manski's maximum score estimator," Economics Letters, Elsevier, vol. 73(2), pages 241-250, November.
    3. Patra, Rohit Kumar & Seijo, Emilio & Sen, Bodhisattva, 2018. "A consistent bootstrap procedure for the maximum score estimator," Journal of Econometrics, Elsevier, vol. 205(2), pages 488-507.
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