# Covariance structure associated with an equality between two general ridge estimators

## TL;DR

This paper establishes a necessary and sufficient condition for when two general ridge estimators are equal in a linear model, based on the error dispersion matrix, extending classical results on estimator equality.

## Contribution

It generalizes existing theorems by characterizing the covariance structure that ensures the equality of two broad classes of ridge estimators.

## Key findings

- Derived a condition based on the dispersion matrix for estimator equality
- Extended classical theorems to a broader class of estimators
- Explored related problems on residual sums of squares and matrix classification

## Abstract

In a general linear model, this paper derives a necessary and sufficient condition under which two general ridge estimators coincide with each other. The condition is given as a structure of the dispersion matrix of the error term. Since the class of estimators considered here contains linear unbiased estimators such as the ordinary least squares estimator and the best linear unbiased estimator, our result can be viewed as a generalization of the well-known theorems on the equality between these two estimators, which have been fully studied in the literature. Two related problems are also considered: equality between two residual sums of squares, and classification of dispersion matrices by a perturbation approach.