Breakdown points of t-type regression estimators

SUMMARY To bound the influence of a leverage point, generalised M-estimators have been suggested. However, the usual generalised M-estimator of regression has a breakdown point that is less than the inverse of its dimension. This paper shows that dimension-independent positive breakdown points can be attained by a class of well-defined generalised Mestimators with redescending scores. The solution can be determined through optimisation of t-type likelihood applied to properly weighted residuals. The highest breakdown point of ' is attained by Cauchy score. These bounded-influence and high-breakdown estimators can be viewed as a fully iterated version of the one-step generalised M-estimates of Simpson, Ruppert & Carroll (1992) with the two advantages of easier interpretability and avoidance of undesirable roots to estimating equations. Given the design-dependent weights, they can be computed via EM algorithms. Empirical investigations show that they are highly competitive with other robust estimators of regression.

[1]  P. Albert,et al.  Models for longitudinal data: a generalized estimating equation approach. , 1988, Biometrics.

[2]  D. G. Simpson,et al.  On One-Step GM Estimates and Stability of Inferences in Linear Regression , 1992 .

[3]  S. Morgenthaler.  Small sample efficiency and exact fit for Cauchy regression models , 1994 .

[4]  Peter J. Rousseeuw,et al.  Robust regression and outlier detection , 1987 .

[5]  Q. Shao,et al.  A general bahadur representation of M-estimators and its application to linear regression with nonstochastic designs , 1996 .

[6]  P. Rousseeuw.  Multivariate estimation with high breakdown point , 1985 .

[7]  David E. Tyler,et al.  Constrained M-Estimation for Regression , 1996 .

[8]  P. Rousseeuw.  Least Median of Squares Regression , 1984 .

[9]  Jeremy MG Taylor,et al.  Robust Statistical Modeling Using the t Distribution , 1989 .

[10]  P. J. Huber.  Finite Sample Breakdown of $M$- and $P$-Estimators , 1984 .

[11]  P. J. Huber.  Robust Estimation of a Location Parameter , 1964 .

[12]  David E. Tyler,et al.  A curious likelihood identity for the multivariate t-distribution , 1994 .

[13]  Werner A. Stahel,et al.  Robust Statistics: The Approach Based on Influence Functions , 1987 .

[14]  Donald B. Rubin,et al.  Max-imum Likelihood from Incomplete Data , 1972 .

[15]  Victor J. Yohai,et al.  Functional stability of one-step GM-estimators in approximately linear regression , 1998 .

[16]  Xuming He.  A local breakdown property of robust tests in linear regression , 1991 .

[17]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[18]  Xuming He.  TAIL BEHAVIOR OF REGRESSION ESTIMATORS AND THEIR BREAKDOWN POINTS , 1990 .

[19]  David E. Tyler.  Finite Sample Breakdown Points of Projection Based Multivariate Location and Scatter Statistics , 1994 .

[20]  R. Maronna.  Robust $M$-Estimators of Multivariate Location and Scatter , 1976 .

[21]  G. Styan,et al.  On the Existence and Uniqueness of the Maximum Likelihood Estimate of a Vector-Valued Parameter in Fixed-Size Samples , 1981 .

[22]  D. G. Simpson,et al.  Reweighting approximate GM estimators: asymptotics and residual-based graphics , 1997 .

[23]  V. Yohai,et al.  Asymptotic behavior of general M-estimates for regression and scale with random carriers , 1981 .