\(M\)-estimates of regression when the scale is unknown and the error distribution is possibly asymmetric: a minimax result. (English) Zbl 1066.62510
Summary: Huber (1964) found the minimax-variance M-estimate of location under the assumption that the scale parameter is known; Li and Zamar (1991) extended this result to the case when the scale is unknown. We consider the robust estimation of the regression coefficients (\(\beta_1\), ..., \(\beta_p\)) when the scale and intercept parameters are unknown. The minimax-variance estimates of (\(\beta_1\), ..., \(\beta_p\)) with respect to the trace of their asymptotic covariance matrix are derived. The maximum is taken over \(\varepsilon\)-contamination neighbourhoods of a central regression model with Gaussian errors (asymmetric contamination is allowed), and the minimum is taken over a large class of generalized M-estimates of regression of the Mallow type. The optimal choice of estimates for the nuisance parameters (scale and intercept) is also considered.
MSC:
| 62F10 | Point estimation |
| 62J05 | Linear regression; mixed models |
| 62F35 | Robustness and adaptive procedures (parametric inference) |
| 62C20 | Minimax procedures in statistical decision theory |
References:
| [1] | Carroll, On estimating variances of robust estimators when the errors are asymmetric, J. Am. Statist. Assoc. 74 pp 674– (1979) · Zbl 0419.62035 |
| [2] | Carroll, A note on asymmetry and robustness in linear regression, Amer. Statist. 42 pp 285– (1988) |
| [3] | Collins, Robust estimation of a location parameter in the presence of asymmetry, Ann. Statist. 4 pp 68– (1976) · Zbl 0351.62035 |
| [4] | Hampel, F.R. (1968). Contributions to the theory of robust estimation. Ph.D. Thesis, Department of Statistics, University of California, Berkeley. |
| [5] | Huber, Robust estimation of a location parameter, Ann. Math. Statist. 36 pp 1753– (1964) · Zbl 0136.39805 |
| [6] | Huber, Robust Statistics (1981) |
| [7] | Li, Min-max asymptotic variance of M-estimates of location when scale is unknown, Statist. Probab. Lett. 11 pp 139– (1991) · Zbl 0733.62031 |
| [8] | Martin, Min-max bias robust regression, Ann. Statist. 17 pp 1608– (1989) · Zbl 0713.62068 |
| [9] | Martin, Bias-robust estimates of scale, Ann. Statist. 21 pp 991– (1993) |
| [10] | Maronna, Asymptotic behavior of general M-estimates for regression and scale with random carriers, Z. Wahrsch. Verw. Geb. 58 pp 7– (1981) · Zbl 0451.62031 |
| [11] | Rousseeuw, Robust and Nonlinear Time Series Analysis pp 256– (1984) · doi:10.1007/978-1-4615-7821-5_15 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
