Bootstrapping M-Estimators of a Multiple Linear Regression Parameter

Consider a multiple linear regression model Yᵢ = x'ᵢβ + εᵢ, where the εᵢ's are independent random variables with common distribution F and the xᵢ's are known design vectors. Let βₙ be the M-estimator of β corresponding to a score function ψ. Under some conditions on F, ψ and the xᵢ's, two-term Edgeworth expansions for the distributions of standardized and studentized βₙ are obtained. Furthermore, it is shown that the bootstrap method is second order correct in the studentized case when the bootstrap samples are drawn from some suitable weighted empirical distribution or from the ordinary empirical distribution of the residuals.