Robust regression with both continuous and categorical predictors.(English)Zbl 0954.62030

Summary: We deal with robust estimation in linear models of the form: $y_i=x_{1i}' \beta_1+ x_{2i}' \beta_2+ e_i\quad (i=1, \dots,n),$ in which the $$x_{1i}$$ are fixed 0-1 vectors – such as an ANOVA design – and the $$x_{2i}$$ are continuous random variables which may contain leverage points. Here $$M$$ estimates are not robust, and $$S$$ estimates may be too expensive. We propose two types of estimates: one is a weighted $$L_1$$ estimate, and the other a combination of $$M$$ and $$S$$ estimates, which attains the maximum breakdown point. The consistency and asymptotic normality of both types of estimates are proved. Simulations suggest that the former is better when the dimension of $$x_{2i}$$ is $$\leq 3$$, and the latter when it is $$\geq 4$$, especially for high contamination.

MSC:

 62F35 Robustness and adaptive procedures (parametric inference) 62J05 Linear regression; mixed models
Full Text:

References:

 [1] Agulló, J., 1996. Computation of estimates with high breakdown point. Ph.D. Thesis, Faculty of Economics, University of Alicante (in Spanish). [2] Barrodale, I.; Roberts, F.D.K., An improved algorithm for discrete l1 linear approximation, SIAM J. numer. anal., 10, 839-848, (1973) · Zbl 0266.65016 [3] Donoho, D.L., Huber, P.J., 1983. The notion of breakdown point. In: A Festschrift for Erich L. Lehmann. Wadsworth, Belmont, CA, pp. 157-184. · Zbl 0523.62032 [4] Ellis, S.P.; Morgenthaler, S., Leverage and breakdown in L1 regression, J. amer. statist. assoc., 87, 143-148, (1992) · Zbl 0781.62101 [5] Hampel, F., Ronchetti, E., Rouseeuw, P.J., Stahel, W.A., 1986. Robust Statistics: The Approach Based on Influence Functions. Wiley, New York. [6] Hawkins, D.M., The feasible set algorithm for least Median of squares regression, Comput. statist. data anal., 16, 81-101, (1993) · Zbl 0875.62305 [7] Hoaglin, D.C., Mosteller, F., Tukey, J.W., 1983. Understanding Robust and Exploratory Data Analysis. Wiley, New York. · Zbl 0599.62007 [8] Huber, P.J., 1981. Robust Statistics. Wiley, New York. · Zbl 0536.62025 [9] Hubert, M., The breakdown value of the L1 estimator in contingency tables, Stat. prob. lett., 33, 419-425, (1997) · Zbl 0899.62073 [10] Hubert, M.; Rousseeuw, P.J., Robust regression with a categorical covariable., (), 215-224 · Zbl 0839.62032 [11] Hubert, M.; Rousseeuw, P.J., Robust regression with both continuous and binary regressors, J. statist. plan. inference, 57, 153-163, (1997) · Zbl 0900.62174 [12] Maronna, R.A.; Bustos, O.H.; Yohai, V.J., Bias- and efficiency-robustness of generalized M-estimators of regression with random carriers., (), 91-116 [13] Maronna, R.A.; Yohai, V.J., The behavior of the stahel-donoho robust multivariate estimators, J. amer. statist. assoc., 90, 330-341, (1995) · Zbl 0820.62050 [14] Maronna, R.A., Yohai, V.J., 1999. Robust regression with both continuous and categorical predictors. Technical Report, Faculty of Exact Sciences, University of Buenos Aires. · Zbl 0954.62030 [15] Mili, L., Coakley, C.W., 1993. Robust estimation in structured linear systems. Technical Report No. 9-13, Department of Statistics, Virginia Polytechnic and State University, Blacksburg. · Zbl 0867.62040 [16] Mizera, I., Müller, C., 1996. Breakdown points and variation exponents of robust M-estimators in linear models, Preprint No. A-22-96, Freie Universität Berlin, Fachbereich Mathematik und Informatik. [17] Müller, C., Breakdown point of designed experiments, J. statist. plan. inference, 45, 413-427, (1995) · Zbl 0827.62066 [18] Rousseeuw, P.J., Leroy, A.M., 1987. Robust Regression and Outlier Detection, Wiley, New York. · Zbl 0711.62030 [19] Rousseeuw, P.J.; Wagner, J., Robust regression with a distributed intercept using least Median of squares, Comput. statist. data anal., 17, 65-76, (1994) · Zbl 0937.62526 [20] Simpson, D.G.; Yohai, V.J., Functional stability of one-step GM-estimators in approximately linear regression, Ann. statist., 26, 1147-1169, (1998) · Zbl 0930.62030 [21] Stahel, W.A., 1981. Breakdown of covariance estimators. Research Report No. 31, Fachgruppe für Statistik, E.T.H. Zürich. [22] Stahel, W.A.; Ruckstuhl, A.F.; Dressler, K., Robust estimation in the analysis of complex molecular spectra, J. amer. statist. assoc., 89, 788-795, (1994) [23] Stromberg, A.J., Computing the exact least Median of squares estimate and stability diagnostics in multiple linear regression, SIAM J. sci. comp., 14, 1289-1299, (1993) · Zbl 0788.65144 [24] Tyler, D., Finite sample breakdown points of projection-based multivariate location and scatter estimates, Ann. statist., 22, 1024-1044, (1994) · Zbl 0815.62015 [25] Yohai, V.J.; Maronna, R.A., Asymptotic behavior of M-estimators for the linear model, Ann. statist., 7, 258-268, (1979) · Zbl 0408.62027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.