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

We consider the (possibly nonlinear) regression model in (mathbb{R }^q) with shift parameter (alpha ) in (mathbb{R }^q) and other parameters (beta ) in (mathbb{R }^p) . Residuals are assumed to be from an unknown distribution function (d.f.). Let (widehat{phi }) be a smooth (M) -estimator of (phi = {{beta }atopwithdelims (){alpha }}) and (T(phi )) a smooth function. We obtain the asymptotic normality, covariance, bias and skewness of (T(widehat{phi })) and an estimator of (T(phi )) with bias (sim n^{-2}) requiring (sim n) calculations. (In contrast, the jackknife and bootstrap estimators require (sim n^2) calculations.) For a linear regression with random covariates of low skewness, if (T(phi ) = nu beta ) , then (T(widehat{phi })) has bias (sim n^{-2}) (not (n^{-1}) ) and skewness (sim n^{-3}) (not (n^{-2}) ), and the usual approximate one-sided confidence interval (CI) for (T(phi )) has error (sim n^{-1}) (not (n^{-1/2}) ). These results extend to random covariates.