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M ‐Estimates of regression when the scale is unknown and the error distribution is possibly asymmetric: A minimax result
Author(s) -
Li Bing,
Zamar Ruben H.
Publication year - 1996
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315625
Subject(s) - minimax , mathematics , statistics , scale (ratio) , covariance , covariance matrix , regression , scale parameter , variance (accounting) , distribution (mathematics) , linear regression , regression analysis , gaussian , mathematical optimization , mathematical analysis , geography , business , physics , cartography , accounting , quantum mechanics
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 ϵ‐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.