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Maximum bias curves for robust regression with non-elliptical regressors. (English) Zbl 1029.62028

Summary: Maximum bias curves for some regression estimates were previously derived assuming that (i) the intercept term is known and/or (ii) the regressors have an elliptical distribution. We present a single method to obtain the maximum bias curves for a large class of regression estimates. Our results are derived under very mild conditions and, in particular, do not require the restrictive assumptions (i) and (ii) above.
Using these results it is shown that the maximum bias curves heavily depend on the shape of the regressors’ distribution which we call the x-configuration. Despite this big effect, the relative performance of different estimates remains unchanged under different x-configurations. We also explore the links between maxbias curves and bias bounds. Finally, we compare the robustness properties of some estimates for the intercept parameter.

MSC:

62F35 Robustness and adaptive procedures (parametric inference)
62J05 Linear regression; mixed models
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[1] Adrover, J. G. (1998). Minimax bias robust estimation of the dispersion matrix of a multivariate distribution. Ann. Statist. 26 2301-2320. · Zbl 0927.62054
[2] Adrover, J. G., Salibian, M. and Zamar, R. H. (1999). Robust inference for the simple linear regression model. Unpublished manuscript. Available at ftp://hajek.stat.ubc.ca/ pub/ruben/rislrm. · Zbl 1032.62027
[3] Berrendero, J. R. and Zamar, R. H. (1995). On the maxbias curve of residual admissible robust regression estimates. Working Paper 95-62, Univ. Carlos III de Madrid. Berrendero, J. R. and Romo, J. (1998) Stabilityunder contamination of robust regression estimates based on differences of residuals. J. Statist. Plann. Inference 70 149-165. · Zbl 1067.62524
[4] Croux, C., Rousseeuw, P. J. and H össjer, O. (1994). Generalized S-estimators. J. Amer. Statist. Assoc. 89 1271-1281. · Zbl 0815.62012
[5] Croux, C., Rousseeuw, P. J. and Van Bael, A. (1996). Positive-breakdown regression byminimizing nested scale estimators. J. Statist. Plann. Inference 53 197-235. · Zbl 0854.62027
[6] Donoho, D. L. and Liu, R. C. (1988). The ”automatic” robustness of minimum distance estimators. Ann. Statist. 16 552-586. · Zbl 0684.62030
[7] Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust Statistics: the Approach based on Influence Functions. Wiley, New York. · Zbl 0593.62027
[8] He, X. (1990). A local breakdown propertyof robust tests in linear regression. J. Multivariate Anal. 38 293-305. · Zbl 0736.62028
[9] He, X. and Simpson, D. G. (1993). Lower bounds for contamination bias: globallyminimax versus locallylinear estimation. Ann. Statist. 21 314-337. · Zbl 0770.62023
[10] Hennig, C. (1995). Efficient high breakdown point estimators in robust regression: which function to choose? Statist. Decisions 13 221-241. · Zbl 0844.62026
[11] H össjer, O. (1992). On the optimalityof S-estimators. Statist. Probab. Lett. 14 413-419. · Zbl 0761.62036
[12] H össjer, O. (1994). Rank-based estimates in the linear model with high breakdown point. J. Amer. Statist. Assoc. 89 149-158. JSTOR: · Zbl 0795.62062
[13] Huber, P. J. (1964). Robust estimation of a location parameter. Ann. Math. Statist. 35 73-101. · Zbl 0136.39805
[14] Huber, P. J. (1981). Robust Statistics. Wiley, New York. · Zbl 0536.62025
[15] Li, G. and Chen, Z. (1985). Projection-pursuit approach to robust dispersion matrices and principal components: primarytheoryand Monte Carlo. J. Amer. Statist. Assoc. 80 759-766. · Zbl 0595.62060
[16] Maronna, R. A. and Yohai, V. J. (1993). Bias-robust estimates of regression based on projections. Ann. Statist. 21 965-990. · Zbl 0787.62037
[17] Martin, R. D., Yohai, V. J. and Zamar, R. H. (1989). Min-max bias robust regression. Ann. Statist. 17 1608-1630. · Zbl 0713.62068
[18] Martin, R. D. and Zamar, R. H. (1989). Asymptotically min-max robust M-estimates of scale for positive random variables. J. Amer. Statist. Assoc. 84 494-501. Martin, R. D. and Zamar, R. (1993) Bias robust estimation of scale. Ann. Statist. 21 991-1017. JSTOR: · Zbl 0787.62038
[19] Rousseeuw, P. J. (1984). Least median of squares regression. J. Amer. Statist. Assoc. 79 871-880. JSTOR: · Zbl 0547.62046
[20] Rousseeuw, P. J. and Hubert, M. (1999). Regression depth. J. Amer. Statist. Assoc. 94 388-402. JSTOR: · Zbl 1007.62060
[21] Rousseeuw, P. J. and Yohai, V. J. (1984). Robust regression bymeans of S-estimators. Robust and Nonlinear Time Series Analysis. Lecture Notes in Statist. 26 256-272. Springer, New York, · Zbl 0567.62027
[22] Simpson, D. G., Ruppert, D. and Carroll, R. J. (1992). On one-step GM-estimates and stability of inferences in linear regression model. J. Amer. Statist. Assoc. 87 439-450. JSTOR: · Zbl 0781.62104
[23] Simpson, D. G. and Yohai, V. J. (1998). Functional stabilityof one-step GM-estimators in linear regression. Ann. Statist. 26 1147-1169. · Zbl 0930.62030
[24] Yohai, V. J. and Zamar, R. H. (1988). High breakdown point estimates of regression bymeans of the minimization of an efficient scale. J. Amer. Statist. Assoc. 83 406-413. JSTOR: · Zbl 0648.62036
[25] Yohai, V. J. and Zamar, R. H. (1993). A minimax-bias propertyof the least -quantile estimates. Ann. Statist. 21 1824-1842. · Zbl 0797.62027
[26] Yohai, V. J. and Zamar, R. H. (1997). Optimal locallyrobust M-estimates of regression. J. Statist. Plann. Inference 64 309-323. · Zbl 0914.62024
[27] Zamar, R. H. (1992). Bias robust estimation in orthogonal regression. Ann. Statist. 20 1875-1888. · Zbl 0784.62051
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