Equality between two general ridge estimators and equivalence of their residual sums of squares

TL;DR

This paper establishes conditions under which two general ridge estimators are equal and when their residual sums of squares are equivalent, expanding understanding of their properties in linear models.

Contribution

It introduces new necessary and sufficient conditions for the equality of two general ridge estimators and their residual sums of squares, including novel column space and bias-based criteria.

Findings

Derived conditions for estimator equality involving column space relationships.

Established criteria for residual sums of squares equivalence.

Extended previous results with new necessary and sufficient conditions.

Abstract

General ridge estimators are typical linear estimators in a general linear model. The class of them includes some shrinkage estimators in addition to classical linear unbiased estimators such as the ordinary least squares estimator and the weighted least squares estimator. We derive necessary and sufficient conditions under which two general ridge estimators coincide. In particular, two noteworthy conditions are added to those from previous studies. The first condition is given as a seemingly column space relationship to the covariance matrix of the error term, and the second one is based on the biases of general ridge estimators. Another problem studied in this paper is to derive an equivalence condition such that equality between two residual sums of squares holds when general ridge estimators are considered.