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

Hirai Mukasa

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.

Equality between two general ridge estimators and applications in several linear 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.
Paper Structure (8 sections, 8 theorems, 55 equations)

This paper contains 8 sections, 8 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Applications
  4. Proofs
  5. Proof of Corollary \ref{['cor1']}
  6. Proof of Corollary \ref{['cor2']}
  7. Proof of Corollary \ref{['cor3']}
  8. Proof of Corollary \ref{['cor4']}

Key Result

Theorem 2.1

The equality G2E holds if and only if $\boldsymbol{\Omega}$ is of the form for some $\boldsymbol{\Gamma} \in {\mathcal{S}} ^+(k)$ and $\boldsymbol{\Delta} \in {\mathcal{S}} ^+(n-k)$ satisfying where $\boldsymbol{Z} \in \mathbb{R}^{n \times (n-k)}$ is any matrix such that $\boldsymbol{X}^\top \boldsymbol{Z} = \boldsymbol{0}$ and ${\mathrm{rank}} (\boldsymbol{Z}) = n-k$.

Theorems & Definitions (23)

  • Theorem 2.1: RefTK20
  • Remark 1
  • Theorem 2.2: RefMT25
  • Remark 2
  • Example 1: Rao's mixed-effects model
  • Corollary 3.1
  • Remark 3
  • Example 2: 2-equation SUR model
  • Corollary 3.2
  • Remark 4
  • ...and 13 more