Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Matias D. Cattaneo, Gregory Fletcher Cox, Michael Jansson, Kenichi Nagasawa

TL;DR

This paper addresses when the non-Gaussian limiting distribution of cube-root or Chernoff-type extremum estimators, represented as $r_n(\hat{\boldsymbol{\theta}}_n-\boldsymbol{\theta}_0) \rightsquigarrow \arg\max_{\mathbf{s}}\mathcal{G}(\mathbf{s})$, possesses a continuous distribution function $F_{\hat{\mathbf{s}}}$. It provides high-level sufficient conditions, centered on a Cameron–Martin RKHS assumption for the mean function of $\mathcal{G}$ and a shift-equivariance property of its covariance, to guarantee continuity of $F_{\hat{\mathbf{s}}}$ in any dimension and for non-bilinear covariances. The authors verify these conditions in three motivating examples—maximum score, empirical risk minimization, and threshold regression—thus supporting bootstrap and related inference procedures that rely on a continuous limit. The work clarifies the distributional properties of the argmax of Gaussian processes in multidimensional settings and discusses the implications for inference when the covariance kernel is not bilinear. Overall, the paper extends the theory of non-Gaussian limits by establishing continuity in a broad, practically relevant class of models and by connecting RKHS structure to the asymptotic distribution of estimators.

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.

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

TL;DR

This paper addresses when the non-Gaussian limiting distribution of cube-root or Chernoff-type extremum estimators, represented as , possesses a continuous distribution function . It provides high-level sufficient conditions, centered on a Cameron–Martin RKHS assumption for the mean function of and a shift-equivariance property of its covariance, to guarantee continuity of in any dimension and for non-bilinear covariances. The authors verify these conditions in three motivating examples—maximum score, empirical risk minimization, and threshold regression—thus supporting bootstrap and related inference procedures that rely on a continuous limit. The work clarifies the distributional properties of the argmax of Gaussian processes in multidimensional settings and discusses the implications for inference when the covariance kernel is not bilinear. Overall, the paper extends the theory of non-Gaussian limits by establishing continuity in a broad, practically relevant class of models and by connecting RKHS structure to the asymptotic distribution of estimators.

Abstract

Certain extremum estimators have asymptotic distributions that are non-Gaussian, yet characterizable as the distribution of the 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.
Paper Structure (17 sections, 2 theorems, 58 equations)

This paper contains 17 sections, 2 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Motivating Examples
  3. Maximum Score
  4. Empirical Risk Minimization
  5. Threshold Regression
  6. Main Result
  7. Verification of Assumption \ref{['Assumption']}
  8. Assumption \ref{['Assumption']}\ref{['Assumption: Existence of argmax']}
  9. Assumption \ref{['Assumption']}\ref{['Assumption: Shift equivariance']}
  10. Assumption \ref{['Assumption']}\ref{['Assumption: RKHS']}
  11. General Strategy
  12. Example: Maximum Score (Continued)
  13. Example: Empirical Risk Minimization (Continued)
  14. Example: Threshold Regression (Continued)
  15. Discussion of Assumption \ref{['Assumption']}\ref{['Assumption: RKHS']}
  16. ...and 2 more sections

Key Result

Lemma 1

Suppose $\mathcal{G}$ has continuous sample paths and suppose Assumption AssumptionAssumption: Shift equivariance holds. For any $\mathbf{h}\in\mathbb{R}^d$, any measurable set $T\subseteq\mathbb{R}^d$, and any compact set $S\subset\mathbb{R}^d$,

Theorems & Definitions (2)

  • Lemma 1
  • Theorem 1