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Consistency of M-estimators for non-identically distributed data: the case of fixed-design distributional regression

Axel Bücher, Johan Segers, Torben Staud

TL;DR

This work extends M-estimator consistency theory to non-identically distributed data via a robust triangular-array framework in fixed-design distributional regression. It provides primitive, model-friendly conditions (notably L^2-dominance and uniform integrability) to guarantee strong and weak consistency of estimators defined by maximizing empirical criteria, even when the criterion may take values in $[-\infty, \infty)$. The authors apply the results to optimum-score estimators, conditional maximum likelihood, and heavy-tailed block-maxima regression, including covariate-dependent Fréchet and GP models, and develop argmax theorems that do not require uniform convergence. These contributions address challenges in extreme-value statistics with parameter-dependent supports and non-random covariates, and are accompanied by detailed proofs and auxiliary lemmas to underpin the results.

Abstract

This paper explores strong and weak consistency of M-estimators for non-identically distributed data, extending prior work. Emphasis is given to scenarios where data is viewed as a triangular array, which encompasses distributional regression models with non-random covariates. Primitive conditions are established for specific applications, such as estimation based on minimizing empirical proper scoring rules or conditional maximum likelihood. A key motivation is addressing challenges in extreme value statistics, where parameter-dependent supports can cause criterion functions to attain the value $-\infty$, hindering the application of existing theorems.

Consistency of M-estimators for non-identically distributed data: the case of fixed-design distributional regression

TL;DR

This work extends M-estimator consistency theory to non-identically distributed data via a robust triangular-array framework in fixed-design distributional regression. It provides primitive, model-friendly conditions (notably L^2-dominance and uniform integrability) to guarantee strong and weak consistency of estimators defined by maximizing empirical criteria, even when the criterion may take values in . The authors apply the results to optimum-score estimators, conditional maximum likelihood, and heavy-tailed block-maxima regression, including covariate-dependent Fréchet and GP models, and develop argmax theorems that do not require uniform convergence. These contributions address challenges in extreme-value statistics with parameter-dependent supports and non-random covariates, and are accompanied by detailed proofs and auxiliary lemmas to underpin the results.

Abstract

This paper explores strong and weak consistency of M-estimators for non-identically distributed data, extending prior work. Emphasis is given to scenarios where data is viewed as a triangular array, which encompasses distributional regression models with non-random covariates. Primitive conditions are established for specific applications, such as estimation based on minimizing empirical proper scoring rules or conditional maximum likelihood. A key motivation is addressing challenges in extreme value statistics, where parameter-dependent supports can cause criterion functions to attain the value , hindering the application of existing theorems.

Paper Structure

This paper contains 13 sections, 21 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Consistency of M-estimators in triangular arrays
  3. Consistency of M-estimators in conditional models with non-random covariates
  4. Applications to optimum score and conditional maximum likelihood estimators
  5. Optimum score estimators with non-random covariates
  6. Conditional maximum likelihood estimation
  7. Fitting heavy-tailed regression models to block maxima
  8. Argmax theorems without uniform convergence
  9. Auxiliary results and proofs
  10. Proofs for Section \ref{['sec:consistency-general']}
  11. Proofs for Section \ref{['sec:consistency-covariates']}
  12. Proofs for Section \ref{['sec:applications']}
  13. Proofs for Section \ref{['sec:argmax']}

Key Result

Theorem 2.1

If Assumptions ass:dgpZ, ass:critZ, ass:trueZ and ass:stochdomZ hold, then every estimator sequence $\hat{\eta}_n$ defined on the same probability space as the array $(Z_{n,i})_{n,i}$ that satisfies $M_n(\hat{\eta}_n) \ge M_n(\eta_0) - \operatorname{o}(1)$ almost surely as $n\to\infty$ is strongly c

Theorems & Definitions (49)

  • Theorem 2.1: Strong consistency
  • Theorem 2.2: Weak consistency
  • Remark 2.3
  • Lemma 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Theorem 4.1
  • Example 4.2: Optimum energy score estimation
  • Corollary 4.3
  • Example 4.4: Generalized extreme value distribution with covariate-dependent parameters
  • ...and 39 more