A GENERAL BAHADUR REPRESENTATION OF M-ESTIMATORS AND ITS APPLICATION TO LINEAR-REGRESSION WITH NONSTOCHASTIC DESIGNS

We obtain strong Bahadur representations for a general class of M-esti mators that satisfies Sigma(i) psi(x(i), theta) = o(delta(n)), where t he x(i)'s are independent but not necessarily identically distributed random variables. The results apply readily to M-estimators of regress ion with nonstochastic designs. More specifically, we consider the min imum L(p) distance estimators, bounded influence GM-estimators and reg ression quantiles. Under appropriate design conditions, the error rate s obtained for the first-order approximations are sharp in these cases . We also provide weaker and more easily verifiable conditions that su ffice for an error rate that is suboptimal but strong enough for deriv ing the asymptotic distribution of Ill-estimators in a wide variety of problems.