Generalized Resubstitution for Regression Error Estimation

TL;DR

This paper introduces generalized resubstitution estimators for regression error that improve bias and variance properties, with proven consistency and demonstrated empirical advantages over traditional methods.

Contribution

It develops a broad family of regression error estimators based on different empirical measures, extending beyond the standard sum of squares, with theoretical guarantees and practical improvements.

Findings

Proposed estimators show superior bias and variance in finite samples.

The estimators are proven to be consistent under broad conditions.

Experimental results confirm improved performance over traditional methods.

Abstract

We propose generalized resubstitution error estimators for regression, a broad family of estimators, each corresponding to a choice of empirical probability measures and loss function. The usual sum of squares criterion is a special case corresponding to the standard empirical probability measure and the quadratic loss. Other choices of empirical probability measure lead to more general estimators with superior bias and variance properties. We prove that these error estimators are consistent under broad assumptions. In addition, procedures for choosing the empirical measure based on the method of moments and maximum pseudo-likelihood are proposed and investigated. Detailed experimental results using polynomial regression demonstrate empirically the superior finite-sample bias and variance properties of the proposed estimators. The R code for the experiments is provided.