Valid edgeworth expansions of M-estimators in regression models with weakly dependent residuals

Consider a linear regression model y(t) = x(t) beta + u(t), where the u(t)'s are weakly dependent random variables, the x(t)'s are known design nonrandom variables, and beta is an unknown parameter. We define an M-estimator <(beta) over cap(n)> of beta corresponding to a smooth score function. Then, the second-order Edgeworth expansion for <(beta) over cap(n)> is derived. Here we do not assume the normality of {u(t)} and {u(t)} includes the usual ARMA processes. Second, we give the second-order Edgeworth expansion for a transformation T<(beta over cap(n))> of <(beta) over cap(n)>. Then, a sufficient condition for T to extinguish the second-order terms is given. The results are applicable to many statistical problems.