Summary
Let (xini, y i be a sequence of independent identically distributed random variables, where x i ∃R pand y i ∃R, and let θ∃R pbe an unknown vector such that y i =x′ i θ+u i (*), where u i is independent of x i and has distribution function F(u/σ), where σ>0 is an unknown parameter. This paper deals with a general class of M-estimates of regression and scale, (θ *,σ*), defined as solutions of the system:\(\sum\limits_i \phi ({\text{x}}_i ,r_i )x_i = 0,\sum\limits_i \chi (|r_i |) = 0,\), where r= (y i −x i 1θ*/σ)*, with Φ∶ R p ×R→R and χ∶ R→R. This class contains estimators of (θ, σ) proposed by Huber, Mallows and Krasker and Welsch. The consistency and asymptotic normality of the general M-estimators are proved assuming general regularity conditions on Φ and χ and assuming the joint distribution of (x i , y i ) to fulfill the model (*) only approximately.
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Maronna, R.A., Yohai, V.J. Asymptotic behavior of general M-estimates for regression and scale with random carriers. Z. Wahrscheinlichkeitstheorie verw Gebiete 58, 7–20 (1981). https://doi.org/10.1007/BF00536192
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DOI: https://doi.org/10.1007/BF00536192