A zero-estimator approach for estimating the signal level in a high-dimensional model-free setting

TL;DR

This paper introduces a zero-estimator approach to accurately estimate the signal level in high-dimensional, model-free regression settings without assuming sparsity or normality, demonstrating improved finite sample performance.

Contribution

It proposes an unbiased, consistent estimator for signal level and a zero-estimator based algorithm to enhance existing estimators in high-dimensional models.

Findings

The estimator is unbiased and consistent.

Zero-estimator approach improves estimation accuracy.

Finite sample performance is demonstrated via simulations.

Abstract

We study a high-dimensional regression setting under the assumption of known covariate distribution. We aim at estimating the amount of explained variation in the response by the best linear function of the covariates (the signal level). In our setting, neither sparsity of the coefficient vector, nor normality of the covariates or linearity of the conditional expectation are assumed. We present an unbiased and consistent estimator and then improve it by using a zero-estimator approach, where a zero-estimator is a statistic whose expected value is zero. More generally, we present an algorithm based on the zero estimator approach that in principle can improve any given estimator. We study some asymptotic properties of the proposed estimators and demonstrate their finite sample performance in a simulation study.