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

2025-01-22

AI Summary

This paper addresses the continuity of the argmax distribution of Gaussian processes—a nonstandard asymptotic limit arising in maximum score estimation, empirical risk minimization, threshold regression, and related problems—whose continuity lacks systematic theoretical justification. To overcome the limitations of existing conditions, which are either overly restrictive or lack generality, we leverage the Cameron–Martin theorem to establish, for the first time, a set of weak yet nearly optimal higher-order sufficient conditions. These conditions constitute a unified framework for assessing continuity of argmax-type non-Gaussian limiting distributions. We rigorously prove that the asymptotic distribution functions of three major classes of estimators are indeed continuous. This result provides foundational theoretical support for constructing confidence sets, justifying bootstrap validity, and developing novel inferential methods—thereby filling a long-standing gap in the continuity analysis of limiting distributions within nonregular statistical inference.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Establishing conditions for continuity of Gaussian process argmax distributions
• Verifying continuity conditions in maximum score and threshold regression
• Supporting inference procedures requiring continuous distribution functions

Innovation

• Establishes continuity conditions for Gaussian process argmax
• Uses Cameron-Martin theorem for weak assumptions verification
• Validates conditions in maximum score and regression examples