Asymtotics of M-estmation in Non-linear Regression

Consider the standard non-linear regression model y i = g(x i , 0)* i , i = 1, ... ,n where g(x, ) is a continuous function on a bounded closed region X , 0 is the unknown parameter vector in R p, {x 1, x 2, ... , x n } is a deterministic design of experiment and {1, 2, ... , n } is a sequence of independent random variables. This paper establishes the existences of M-estimates and the asymptotic uniform linearity of M-scores in a family of non-linear regression models when the errors are independent and identically distributed. This result is then used to obtain the asymptotic distribution of a class of M-estimators for a large class of non-linear regression models. At the same time, we point out that Theorem 2 of Wang (1995) (J. of Multivariate Analysis, vol. 54, pp. 227238, Corrigenda. vol. 55, p. 350) is not correct.