Asymptotic properties of M-estimators in linear and nonlinear multivariate regression models

We consider the (possibly nonlinear) regression model in with shift parameter in and other parameters in . Residuals are assumed to be from an unknown distribution function (d.f.). Let be a smooth -estimator of and a smooth function. We obtain the asymptotic normality, covariance, bias and skewness of and an estimator of with bias requiring calculations. (In contrast, the jackknife and bootstrap estimators require calculations.) For a linear regression with random covariates of low skewness, if , then has bias (not ) and skewness (not ), and the usual approximate one-sided confidence interval (CI) for has error (not ). These results extend to random covariates.