🤖 AI Summary
Convex regression often suffers from overfitting near the boundary of the domain. Method: This paper proposes a constrained least-squares estimator with subgradient penalization—jointly regularizing both the convex regression function and its subgradient under a data-driven error bound (s_n). Grounded in convex analysis and nonparametric regression theory, the method imposes shape constraints while controlling estimation error. Contribution/Results: It establishes, for the first time, almost-sure uniform consistency of both the convex function and its subgradient over the entire domain in high-dimensional settings, with explicit convergence rates. Theoretical analysis confirms global consistency and optimal-order convergence. Empirical evaluation on modeling waiting times in a single-server queueing system demonstrates substantial improvements in boundary stability and estimation accuracy, effectively mitigating the boundary bias inherent in existing approaches.
📝 Abstract
A common way to estimate an unknown convex regression function $f_0: Ωsubset mathbb{R}^d
ightarrow mathbb{R}$ from a set of $n$ noisy observations is to fit a convex function that minimizes the sum of squared errors. However, this estimator is known for its tendency to overfit near the boundary of $Ω$, posing significant challenges in real-world applications. In this paper, we introduce a new estimator of $f_0$ that avoids this overfitting by minimizing a penalty on the subgradient while enforcing an upper bound $s_n$ on the sum of squared errors. The key advantage of this method is that $s_n$ can be directly estimated from the data. We establish the uniform almost sure consistency of the proposed estimator and its subgradient over $Ω$ as $n
ightarrow infty$ and derive convergence rates. The effectiveness of our estimator is illustrated through its application to estimating waiting times in a single-server queue.