Root n consistent extremile regression and its supervised and semi-supervised learning

TL;DR

This paper introduces a root n-consistent estimator for linear extremile regression and develops a semi-supervised framework utilizing unlabeled data to improve estimation accuracy, with validation through simulations and real data.

Contribution

It proposes a novel root n-consistent estimator for extremile regression and a semi-supervised learning framework incorporating unlabeled data.

Findings

Estimator achieves root n-consistency.

Semi-supervised approach improves estimation accuracy.

Method performs well in simulations and real data.

Abstract

Extremile (Daouia, Gijbels and Stupfler,2019) is a novel and coherent measure of risk, determined by weighted expectations rather than tail probabilities. It finds application in risk management, and, in contrast to quantiles, it fulfills the axioms of consistency, taking into account the severity of tail losses. However, existing studies (Daouia, Gijbels and Stupfler,2019,2022) on extremile involve unknown distribution functions, making it challenging to obtain a root n-consistent estimator for unknown parameters in linear extremile regression. This article introduces a new definition of linear extremile regression and its estimation method, where the estimator is root n-consistent. Additionally, while the analysis of unlabeled data for extremes presents a significant challenge and is currently a topic of great interest in machine learning for various classification problems, we have developed a semi-supervised framework for the proposed extremile regression using unlabeled data. This framework can also enhance estimation accuracy under model misspecification. Both simulations and real data analyses have been conducted to illustrate the finite sample performance of the proposed methods.