M-estimates of regression when the scale is unknown and the error distribution is possibly asymmetric: A minimax result

Bing Li, Ruben H. Zamar

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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 (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.

Original languageEnglish (US)
Pages (from-to)193-206
Number of pages14
JournalCanadian Journal of Statistics
Volume24
Issue number2
DOIs
StatePublished - Jan 1 1996

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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