Equality between two general ridge estimators and applications in several linear models

TL;DR

This paper establishes conditions under which a covariance-free general ridge estimator is equivalent to the traditional one, simplifying estimation in various linear models with unknown error covariance.

Contribution

It derives verifiable conditions for using a covariance-free ridge estimator across multiple linear models, reducing the need for a two-step estimation process.

Findings

Conditions for estimator equivalence are provided for several models.

Covariance-free estimator simplifies the estimation process.

Results applicable to models like mixed-effects, spatial, and serial correlation models.

Abstract

General ridge estimators are widely used in the general linear model because they possess desirable properties such as linear sufficiency and linear admissibility. However, when the covariance matrix of the error term is partially unknown, estimation typically requires a two-step procedure. This paper derives conditions under which the general ridge estimator based on the covariance matrix coincides with the one that does not depend on it. In particular, we provide practically verifiable conditions for several linear models, including Rao's mixed-effects model, a seemingly unrelated regression model, first-order spatial autoregressive and spatial moving average models, and serial correlation models. These results enable the use of a covariance-free general ridge estimator, thereby simplifying the two-step estimation procedure.