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

We consider the (possibly nonlinear) regression model in $$\mathbb{R }^q$$ R q with shift parameter $$\alpha $$ α in $$\mathbb{R }^q$$ R q and other parameters $$\beta $$ β in $$\mathbb{R }^p$$ R p . Residuals are assumed to be from an unknown distribution function (d.f.). Let $$\widehat{\phi }$$ ϕ ^ be a smooth $$M$$ M -estimator of $$\phi={{\beta }\atopwithdelims (){\alpha }}$$ ϕ = β α and $$T(\phi )$$ T ( ϕ ) a smooth function. We obtain the asymptotic normality, covariance, bias and skewness of $$T(\widehat{\phi })$$ T ( ϕ ^ ) and an estimator of $$T(\phi )$$ T ( ϕ ) with bias $$\sim n^{-2}$$ ∼ n - 2 requiring $$\sim n$$ ∼ n calculations. (In contrast, the jackknife and bootstrap estimators require $$\sim n^2$$ ∼ n 2 calculations.) For a linear regression with random covariates of low skewness, if $$T(\phi )=\nu \beta $$ T ( ϕ ) = ν β , then $$T(\widehat{\phi })$$ T ( ϕ ^ ) has bias $$\sim n^{-2}$$ ∼ n - 2 (not $$n^{-1}$$ n - 1 ) and skewness $$\sim n^{-3}$$ ∼ n - 3 (not $$n^{-2}$$ n - 2 ), and the usual approximate one-sided confidence interval (CI) for $$T(\phi )$$ T ( ϕ ) has error $$\sim n^{-1}$$ ∼ n - 1 (not $$n^{-1/2}$$ n - 1 / 2 ). These results extend to random covariates. Copyright Springer-Verlag Berlin Heidelberg 2014