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Estimates of MM type for the multivariate linear model

Nadia L. Kudraszow, Ricardo A. Maronna

TL;DR

The paper extends MM-estimation to multivariate linear models by jointly estimating regression and error-structure parameters with a high breakdown point and strong Gaussian efficiency under elliptical errors. It defines a robust iterative procedure combining a high-breakdown initial estimate with M-estimation of scale and a second-stage minimization, and proves consistency and asymptotic normality under specified conditions. The authors provide an explicit computing algorithm and demonstrate, through simulations and a real data example, that MLM MM-estimates offer favorable robustness-accuracy trade-offs compared to S- and τ-estimates and the MLE, including reliable outlier detection. This work delivers a practical, theoretically sound framework for robust multivariate regression analysis with scalable computation.

Abstract

We propose a class of robust estimates for multivariate linear models. Based on the approach of MM estimation (Yohai 1987), we estimate the regression coefficients and the covariance matrix of the errors simultaneously. These estimates have both high breakdown point and high asymptotic efficiency under Gaussian errors. We prove consistency and asymptotic normality assuming errors with an elliptical distribution. We describe an iterative algorithm for the numerical calculation of these estimates. The advantages of the proposed estimates over their competitors are demonstrated through both simulated and real data.

Estimates of MM type for the multivariate linear model

TL;DR

The paper extends MM-estimation to multivariate linear models by jointly estimating regression and error-structure parameters with a high breakdown point and strong Gaussian efficiency under elliptical errors. It defines a robust iterative procedure combining a high-breakdown initial estimate with M-estimation of scale and a second-stage minimization, and proves consistency and asymptotic normality under specified conditions. The authors provide an explicit computing algorithm and demonstrate, through simulations and a real data example, that MLM MM-estimates offer favorable robustness-accuracy trade-offs compared to S- and τ-estimates and the MLE, including reliable outlier detection. This work delivers a practical, theoretically sound framework for robust multivariate regression analysis with scalable computation.

Abstract

We propose a class of robust estimates for multivariate linear models. Based on the approach of MM estimation (Yohai 1987), we estimate the regression coefficients and the covariance matrix of the errors simultaneously. These estimates have both high breakdown point and high asymptotic efficiency under Gaussian errors. We prove consistency and asymptotic normality assuming errors with an elliptical distribution. We describe an iterative algorithm for the numerical calculation of these estimates. The advantages of the proposed estimates over their competitors are demonstrated through both simulated and real data.

Paper Structure

This paper contains 14 sections, 20 theorems, 192 equations, 8 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Definition and properties
  3. Breakdown Point
  4. Influence function
  5. Consistency
  6. Asymptotic Normality
  7. Computing algorithm
  8. Simulation
  9. Simulation design
  10. Description of the estimators
  11. Results
  12. An example with real data
  13. Conclusions
  14. Appendix

Key Result

Theorem 1

Let $\mathbf{Z}=\{\mathbf{z}_1,\ldots, \mathbf{z}_n\}$ be a sample of size $n$ satisfying the MLM MLM, where $\mathbf{z}_i=(\mathbf{y}_i',\mathbf{x}_i')$. If $k_n/n< 0.5$ then there is a pair $(\widehat{\mathbf{B}}_n,\widehat{\mathbf{\Gamma}}_n)$ that minimizes the function $S(\mathbf{B},\mathbf{\Ga

Figures (8)

  • Figure 1: Simulation: mean squared errors for $q=2$ and $x_0=1$.
  • Figure 2: Simulation: mean squared errors for $q=2$ and $x_0=10$.
  • Figure 3: Simulation: mean squared errors for $q=5$ and $x_0=1$.
  • Figure 4: Simulation: mean squared errors for $q=5$ and $x_0=10$.
  • Figure 5: QQ-plots of the Mahalanobis norms of the residuals of the MM-estimate (left), the MLE (right) and the MM-estimate in the same interval as the MLE (center).
  • ...and 3 more figures

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Remark 3
  • Definition 4
  • Theorem 3
  • ...and 22 more