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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

Certain extremum estimators have asymptotic distributions that are non-Gaussian, yet characterizable as the distribution of 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 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 Cameron-Martin theorem. In a leading special case, the assumption in question is demonstrably weak and appears to be close to minimal.

Suggested Citation

  • 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, revised Oct 2025.
    • Handle: RePEc:arx:papers:2501.13265
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    References listed on IDEAS

    as
    1. Jushan Bai, 1997. "Estimation Of A Change Point In Multiple Regression Models," The Review of Economics and Statistics, MIT Press, vol. 79(4), pages 551-563, November.
    2. Bruce E. Hansen, 2000. "Sample Splitting and Threshold Estimation," Econometrica, Econometric Society, vol. 68(3), pages 575-604, May.
    3. Matias D. Cattaneo & Michael Jansson & Kenichi Nagasawa, 2023. "Bootstrap-Assisted Inference for Generalized Grenander-type Estimators," Papers 2303.13598, arXiv.org, revised Jul 2024.
    4. 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.
    5. Matias D. Cattaneo & Michael Jansson & Kenichi Nagasawa, 2020. "Bootstrap‐Based Inference for Cube Root Asymptotics," Econometrica, Econometric Society, vol. 88(5), pages 2203-2219, September.
    6. Ping Yu & Xiaodong Fan, 2021. "Threshold Regression With a Threshold Boundary," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(4), pages 953-971, October.
    7. Horowitz, Joel L, 1992. "A Smoothed Maximum Score Estimator for the Binary Response Model," Econometrica, Econometric Society, vol. 60(3), pages 505-531, May.
    8. 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.
    9. 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.
    10. Adam M Rosen & Takuya Ura, 2025. "Finite Sample Inference for the Maximum Score Estimand," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 92(6), pages 4117-4151.
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