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Multivariate MM-estimators with auxiliary Scale for Linear Models with Structured Covariance Matrices

Hendrik Paul Lopuhaa

TL;DR

This work develops a unified MM-estimation framework with auxiliary scale for balanced linear models with structured covariance, extending to linear mixed effects and several multivariate models. By introducing MM-functionals based on two losses $\rho_0$ and $\rho_1$ and an auxiliary scale, it establishes existence, continuity, consistency, and asymptotic normality under mild distributional assumptions, far beyond elliptically contoured densities. The authors derive global breakdown points and local influence functions, including explicit forms in the general case and under elliptical models, linking to existing multivariate MM/S-functionals and CM-functionals. A comprehensive application demonstrates practical tuning via Tukey biweight losses, quantifying the trade-off between efficiency and robustness through scalar indices, and providing guidance for robust covariance estimation in complex multivariate settings.

Abstract

We provide a unified approach to MM-estimation with auxiliary scale for balanced linear models with structured covariance matrices. This approach leads to estimators that are highly robust against outliers and highly efficient for normal data. These properties not only hold for estimators of the regression parameter, but also for estimators of scale invariant transformations of the variance parameters. Of main interest are MM-estimators for linear mixed effects models, but our approach also includes MM-estimators in several other standard multivariate models. We provide sufficient conditions for the existence of MM-functionals and MM-estimators, establish asymptotic properties such as consistency and asymptotic normality, and derive their robustness properties in terms of breakdown point and influence function. All the results are obtained for general identifiable covariance structures and are established under mild conditions on the distribution of the observations, which goes far beyond models with elliptically contoured densities.

Multivariate MM-estimators with auxiliary Scale for Linear Models with Structured Covariance Matrices

TL;DR

This work develops a unified MM-estimation framework with auxiliary scale for balanced linear models with structured covariance, extending to linear mixed effects and several multivariate models. By introducing MM-functionals based on two losses and and an auxiliary scale, it establishes existence, continuity, consistency, and asymptotic normality under mild distributional assumptions, far beyond elliptically contoured densities. The authors derive global breakdown points and local influence functions, including explicit forms in the general case and under elliptical models, linking to existing multivariate MM/S-functionals and CM-functionals. A comprehensive application demonstrates practical tuning via Tukey biweight losses, quantifying the trade-off between efficiency and robustness through scalar indices, and providing guidance for robust covariance estimation in complex multivariate settings.

Abstract

We provide a unified approach to MM-estimation with auxiliary scale for balanced linear models with structured covariance matrices. This approach leads to estimators that are highly robust against outliers and highly efficient for normal data. These properties not only hold for estimators of the regression parameter, but also for estimators of scale invariant transformations of the variance parameters. Of main interest are MM-estimators for linear mixed effects models, but our approach also includes MM-estimators in several other standard multivariate models. We provide sufficient conditions for the existence of MM-functionals and MM-estimators, establish asymptotic properties such as consistency and asymptotic normality, and derive their robustness properties in terms of breakdown point and influence function. All the results are obtained for general identifiable covariance structures and are established under mild conditions on the distribution of the observations, which goes far beyond models with elliptically contoured densities.

Paper Structure

This paper contains 23 sections, 30 theorems, 330 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Balanced linear models with structured covariances
  3. Definitions
  4. Existence
  5. Continuity and consistency
  6. Global robustness: the breakdown point
  7. Score equations
  8. Local robustness: the influence function
  9. The general case
  10. Elliptically contoured densities
  11. Asymptotic normality
  12. The general case
  13. Elliptical contoured densities
  14. Application
  15. Supplemental Material
  16. ...and 8 more sections

Key Result

Lemma 1

Let $\rho_0$ satisfy (R1)-(R3) and let $(\bm\beta_0(P),\bm\theta_0(P))\in\mathbb{R}^{q}\times\bm\Theta$ be the pair of initial functionals at $P$, such that (C0) holds. Then a solution $\sigma(P)>0$ to def:sigma exists and is unique.

Figures (2)

  • Figure 1: Asymptotic efficiencies at the multivariate normal distribution (top row) for the MM-estimators of regression (solid) and for shape and direction (dashed) and the GES (bottom row) of the corresponding MM-functionals for dimensions $k=2,5,10$.
  • Figure 2: Asymptotic efficiencies at the multivariate Student distribution with degrees of freedom $\nu=1$ (top row) and $\nu=15$ (bottom row) for the MM-estimators of regression (solid) and for shape and direction (dashed) for dimensions $k=2,5,10$.

Theorems & Definitions (72)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Lemma 1
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • Corollary 3
  • ...and 62 more