arXiv subjects

Eunji Lim

Publications and source records attributed to Eunji Lim.

3 recordsLinked to original sources

arXiv2025

Convex Regression with a Penalty

A common way to estimate an unknown convex regression function $f_0: \Omega \subset \mathbb{R}^d \rightarrow \mathbb{R}$ 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 $\Omega$, 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 $\Omega$ as $n \rightarrow \infty$ and derive convergence rates. The effectiveness of our estimator is illustrated through its application to estimating waiting times in a single-server queue.

stat.ML↗

arXiv2024

Estimating a Function and Its Derivatives Under a Smoothness Condition

We consider the problem of estimating an unknown function f* and its partial derivatives from a noisy data set of n observations, where we make no assumptions about f* except that it is smooth in the sense that it has square integrable partial derivatives of order m. A natural candidate for the estimator of f* in such a case is the best fit to the data set that satisfies a certain smoothness condition. This estimator can be seen as a least squares estimator subject to an upper bound on some measure of smoothness. Another useful estimator is the one that minimizes the degree of smoothness subject to an upper bound on the average of squared errors. We prove that these two estimators are computable as solutions to quadratic programs, establish the consistency of these estimators and their partial derivatives, and study the convergence rate as n increases to infinity. The effectiveness of the estimators is illustrated numerically in a setting where the value of a stock option and its second derivative are estimated as functions of the underlying stock price.

stat.ML↗

arXiv2024

Overfitting Reduction in Convex Regression

Convex regression is a method for estimating the convex function from a data set. This method has played an important role in operations research, economics, machine learning, and many other areas. However, it has been empirically observed that convex regression produces inconsistent estimates of convex functions and extremely large subgradients near the boundary as the sample size increases. In this paper, we provide theoretical evidence of this overfitting behavior. To eliminate this behavior, we propose two new estimators by placing a bound on the subgradients of the convex function. We further show that our proposed estimators can reduce overfitting by proving that they converge to the underlying true convex function and that their subgradients converge to the gradient of the underlying function, both uniformly over the domain with probability one as the sample size is increasing to infinity. An application to Finnish electricity distribution firms confirms the superior performance of the proposed methods in predictive power over the existing methods.

stat.ME↗