BOOTSTRAPPING M-ESTIMATORS OF A MULTIPLE LINEAR-REGRESSION PARAMETER

Consider a multiple linear regression model Y(i) = x(i)'beta + epsilon(i), where the epsilon(i)'s are independent random variables with common distribution F and the x(i)'s are known design vectors. Let Beta(n)BAR be the M-estimator of beta corresponding to a score function psi. Under some conditions on F, psi and the x(i)'s, two-term Edgeworth expansions for the distributions of standardized and studentized beta(n)BAR 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.