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

摘要

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