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

TL;DR

This paper investigates the consistency of M-estimators in non-i.i.d. data scenarios, especially in distributional regression with fixed covariates, addressing challenges like parameter-dependent supports.

Contribution

It extends existing consistency results to non-i.i.d. data viewed as a triangular array, including applications with non-random covariates and extreme value statistics.

Findings

Established primitive conditions for consistency in distributional regression.

Addressed issues with criterion functions in extreme value models.

Extended theoretical results to non-i.i.d. triangular array data.

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.