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Asymptotic behavior of general M-estimates for regression and scale with random carriers

Research paper by Ricardo A. Maronna, Victor J. Yohai

Indexed on: 01 Mar '81Published on: 01 Mar '81Published in: Probability Theory and Related Fields



Abstract

Let (xini, yibe a sequence of independent identically distributed random variables, where xi∃Rpand yi∃R, and let θ∃Rpbe an unknown vector such that yi=x′iθ+ui(*), where uiis independent of xiand 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= (yi−xi1θ*/σ)*, with Φ∶ Rp×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 (xi, yi) to fulfill the model (*) only approximately.