A journey in single steps: robust one-step M-estimation
A journey in single steps: robust one-step M-estimation
We present a unified treatment of different types of one-step M-estimation in regression models which incorporates the Newton–Raphson, method of scoring and iteratively reweighted least squares forms of one-step estimator. We use higher order expansions to distinguish between the different forms of estimator and the effects of different initial estimators. We show that the Newton–Raphson form has better properties than the method of scoring form which, in turn, has better properties than the iteratively reweighted least squares form. We also show that the best choice of initial estimator is a smooth, robust estimator which converges at the rate n?1/2. These results have important consequences for the common data-analytic strategy of using a least squares analysis on "clean" data obtained by deleting observations with extreme residuals from an initial least squares fit. It is shown that the resulting estimator is an iteratively reweighted least squares one-step estimator with least squares as the initial estimator, giving it the worst performance of the one-step estimators we consider: inferences resulting from this strategy are neither valid nor robust.
breakdown point, diagnostics, influence function, iteratively reweighted least squares estimator, M-estimator, method of scoring estimator, Newton–Raphson estimator, outliers, rejection method, S-estimator
287-310
Welsh, A.H.
27640871-afff-4d45-a191-8a72abee4c1a
Ronchetti, Elvezio
7b812071-0e0c-495e-a87b-0fe4d6210437
2002
Welsh, A.H.
27640871-afff-4d45-a191-8a72abee4c1a
Ronchetti, Elvezio
7b812071-0e0c-495e-a87b-0fe4d6210437
Welsh, A.H. and Ronchetti, Elvezio
(2002)
A journey in single steps: robust one-step M-estimation.
Journal of Statistical Planning and Inference, 103 (1-2), .
(doi:10.1016/S0378-3758(01)00228-2).
Abstract
We present a unified treatment of different types of one-step M-estimation in regression models which incorporates the Newton–Raphson, method of scoring and iteratively reweighted least squares forms of one-step estimator. We use higher order expansions to distinguish between the different forms of estimator and the effects of different initial estimators. We show that the Newton–Raphson form has better properties than the method of scoring form which, in turn, has better properties than the iteratively reweighted least squares form. We also show that the best choice of initial estimator is a smooth, robust estimator which converges at the rate n?1/2. These results have important consequences for the common data-analytic strategy of using a least squares analysis on "clean" data obtained by deleting observations with extreme residuals from an initial least squares fit. It is shown that the resulting estimator is an iteratively reweighted least squares one-step estimator with least squares as the initial estimator, giving it the worst performance of the one-step estimators we consider: inferences resulting from this strategy are neither valid nor robust.
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Published date: 2002
Keywords:
breakdown point, diagnostics, influence function, iteratively reweighted least squares estimator, M-estimator, method of scoring estimator, Newton–Raphson estimator, outliers, rejection method, S-estimator
Organisations:
Statistics
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Local EPrints ID: 29942
URI: http://eprints.soton.ac.uk/id/eprint/29942
ISSN: 0378-3758
PURE UUID: 32e0bee8-4a03-4c7d-ab1f-3c58a428b80d
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Date deposited: 10 May 2006
Last modified: 15 Mar 2024 07:36
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Author:
A.H. Welsh
Author:
Elvezio Ronchetti
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