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

TL;DR

This paper establishes conditions under which the asymptotic distribution of the argmax of a Gaussian process is continuous, supporting inference procedures in various estimators like maximum score and threshold regression.

Contribution

It provides high-level sufficient conditions for the continuity of the distribution function of the argmax of Gaussian processes, verified in key econometric examples.

Findings

Conditions enable use of Cameron-Martin theorem

Continuity of distribution function verified in examples

Supports validity of recent inference procedures

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.