Debiased Regression for Root-N-Consistent Conditional Mean Estimation (2411.11748v3)

Abstract: This study introduces a debiasing method for regression estimators, including high-dimensional and nonparametric regression estimators. For example, nonparametric regression methods allow for the estimation of regression functions in a data-driven manner with minimal assumptions; however, these methods typically fail to achieve $\sqrt{n}$-consistency in their convergence rates, and many, including those in machine learning, lack guarantees that their estimators asymptotically follow a normal distribution. To address these challenges, we propose a debiasing technique for nonparametric estimators by adding a bias-correction term to the original estimators, extending the conventional one-step estimator used in semiparametric analysis. Specifically, for each data point, we estimate the conditional expected residual of the original nonparametric estimator, which can, for instance, be computed using kernel (Nadaraya-Watson) regression, and incorporate it as a bias-reduction term. Our theoretical analysis demonstrates that the proposed estimator achieves $\sqrt{n}$-consistency and asymptotic normality under a mild convergence rate condition for both the original nonparametric estimator and the conditional expected residual estimator. Notably, this approach remains model-free as long as the original estimator and the conditional expected residual estimator satisfy the convergence rate condition. The proposed method offers several advantages, including improved estimation accuracy and simplified construction of confidence intervals.

• The paper presents a novel debiasing technique that adds a bias-correction term to achieve √n-consistent and asymptotically normal regression estimation.
• It constructs the correction term by estimating conditional expected residuals with kernel and series regressions, improving overall estimator accuracy.
• The debiased estimator is double robust and semiparametrically efficient, ensuring reliable performance in high-dimensional, real-world applications.

Debiased Regression for Root-N-Consistent Conditional Mean Estimation

The paper under discussion presents a methodology for achieving root-n-consistent estimation in regression analysis. This new technique builds upon nonparametric regression methods, which, while advantageous for their flexibility and minimal assumptions, typically struggle to guarantee $\sqrt{n}$-consistency and asymptotic normality. By introducing a debiasing technique reliant on a bias-correction term, this paper extends traditional one-step estimators from semiparametric analysis to enhance the consistency and accuracy of regression estimators across both high-dimensional and nonparametric contexts.

Key Contributions

1. Debiasing Technique: The paper proposes a debiasing approach where a bias-correction term is added to a base estimator to attain $\sqrt{n}$-consistency and asymptotic normality. This technique is model-free as long as the original regression estimator and the conditional expected residual estimator meet mild convergence rate conditions.
2. Conditional Expected Residuals: The method employs a construction of the bias-correction term by estimating conditional expected residuals using regression techniques like kernel and series regression. This enables accurate bias reduction and consistency in estimation.
3. Efficient Estimator: The debiased estimator achieves semiparametric efficiency because its asymptotic variance aligns with the derived efficiency bound, which is the lower bound for regular, asymptotically linear estimators. This signifies that the proposed method reaches optimal variance performance under local model perturbations.
4. Double Robustness: The estimator remains consistent if either the original regression estimator or the conditional expected residual estimator is consistent. This property is significant in practical applications, providing robustness against model specification errors.

Methodology

The debiased estimator extends beyond traditional parametric restrictions by incorporating methods like Nadaraya-Watson and series regressions to capture conditional expected residuals. The estimator is primarily concerned with correcting the bias that arises due to variances unaccounted for in high-dimensional and nonparametric settings. Additionally, the asymptotic properties arguments in the paper rely on standard statistical techniques such as sample splitting or assuming Donsker conditions to handle empirical process terms.

Theoretical and Practical Implications

The research highlights significant theoretical implications, particularly in the context of nonparametric regression's minimax lower bounds. Unlike the bounds that apply under worst-case scenarios, the paper focuses on achieving asymptotic optimality under the true data generating process with minimal parameter assumptions. Practically, the simpler construction of confidence intervals afforded by the bias correction term, combined with doubled robustness, ensures that the methodology can be directly applied in real-world data scenarios where assumptions of parametric models are not tenable.

Future Directions

Given the debiasing framework's ability to adapt across different regression contexts, future research could explore its integration into more complex machine learning models that often suffer from high bias and variance issues. Additionally, extending this framework to other statistical estimation challenges could offer broad utility across various fields like econometrics and biomedical research, where precise estimation and inference are vital.

In conclusion, the proposed debiasing technique represents a significant advancement in achieving consistent and asymptotically normal estimators in nonparametric and high-dimensional regression. The combination of theoretical soundness, practical applicability, and robustness paves the way for its adoption in diverse complex-data environments where traditional methods may falter.