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

Summary Let (xini, y_ibe a sequence of independent identically distributed random variables, where x_iR^pand y_iR, and let R^pbe an unknown vector such that y_i=x_i+u_i(^*), where u_iis independent of x_iand has distribution function F(u/), where >0 is an unknown parameter. This paper deals with a general class of M-estimates of regression and scale, (^*,^*), defined as solutions of the system:, where r= (y_i–x_i^1*/)^*, with R^p×RR and RR. This class contains estimators of (, ) 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 and and assuming the joint distribution of (x_i, y_i) to fulfill the model (^*) only approximately.