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

• Autores: Maronna, Ricardo Antonio; Yohai, Victor J.
• Año de publicación: 1981
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: Ciencias Exactas; Matemática; M-estimates
• Institución: Universidad Nacional de La Plata

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:∑iϕ(xi,ri)xi=0,∑iχ(|ri|)=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.
Facultad de Ciencias Exactas