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Econometrica Vol. 94 No. 3 2026

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

Matias D. Cattaneo1; Gregory F. Cox2; Michael Jansson3; Kenichi Nagasawa4

1 Department of Operations Research and Financial Engineering, Princeton University · 2 Department of Economics, National University of Singapore · 3 Department of Economics, UC Berkeley and ACE · 4 Department of Economics, University of Warwick

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Abstract

Certain extremum estimators have asymptotic distributions that are non‐Gaussian, yet characterizable as the distribution of the arg max 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.

DOI
10.3982/ecta23862
Volume
94
Issue
3
Pages
941-955
Language
en
Sources
crossref openalex

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