Robust oracle estimation and uncertainty quantification for possibly sparse quantiles

TL;DR

This paper introduces a robust penalization method for high-dimensional quantile estimation that handles non-normal, dependent noise and achieves oracle-like performance in estimation and uncertainty quantification.

Contribution

It develops a novel penalization approach based on quantile loss with adaptive penalties, addressing sparse high-dimensional quantile estimation under general noise conditions.

Findings

Method achieves oracle-like estimation accuracy.

Procedure provides valid uncertainty quantification.

Results are adaptive minimax over sparsity scales.

Abstract

A general many quantiles + noise model is studied in the robust formulation (allowing non-normal, non-independent observations), where the identifiability requirement for the noise is formulated in terms of quantiles rather than the traditional zero expectation assumption. We propose a penalization method based on the quantile loss function with appropriately chosen penalty function making inference on possibly sparse high-dimensional quantile vector. We apply a local approach to address the optimality by comparing procedures to the oracle sparsity structure. We establish that the proposed procedure mimics the oracle in the problems of estimation and uncertainty quantification (under the so called EBR condition). Adaptive minimax results over sparsity scale follow from our local results.