### Abstract

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 (β_{1} , . . . , β_{p}) when the scale and the intercept parameters are unknown. The minimax-variance estimates of (β_{1} , . . . , β_{p}) with respect to the trace of their asymptotic covariance matrix are derived. The maximum is taken over ∈-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.

Original language | English (US) |
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Pages (from-to) | 193-206 |

Number of pages | 14 |

Journal | Canadian Journal of Statistics |

Volume | 24 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1996 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

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*Canadian Journal of Statistics*, vol. 24, no. 2, pp. 193-206. https://doi.org/10.2307/3315625

**M-estimates of regression when the scale is unknown and the error distribution is possibly asymmetric : A minimax result.** / Li, Bing; Zamar, Ruben H.

Research output: Contribution to journal › Article

TY - JOUR

T1 - M-estimates of regression when the scale is unknown and the error distribution is possibly asymmetric

T2 - A minimax result

AU - Li, Bing

AU - Zamar, Ruben H.

PY - 1996/1/1

Y1 - 1996/1/1

N2 - 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 (β1 , . . . , βp) when the scale and the intercept parameters are unknown. The minimax-variance estimates of (β1 , . . . , βp) with respect to the trace of their asymptotic covariance matrix are derived. The maximum is taken over ∈-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.

AB - 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 (β1 , . . . , βp) when the scale and the intercept parameters are unknown. The minimax-variance estimates of (β1 , . . . , βp) with respect to the trace of their asymptotic covariance matrix are derived. The maximum is taken over ∈-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.

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U2 - 10.2307/3315625

DO - 10.2307/3315625

M3 - Article

VL - 24

SP - 193

EP - 206

JO - Canadian Journal of Statistics

JF - Canadian Journal of Statistics

SN - 0319-5724

IS - 2

ER -