Bahadur-Kiefer representations for GM-estimators in autoregression models

AbstractThis paper proves strong consistency, along with a rate, of a class of generalized M-estimators for the autoregression parameter vector in pth order autoregression (AR(p)) models. If the score function ψ has bounded second derivative then the rate of convergence is n−12(lnlnn)12 while for a general ψ it is n−12(lnn)12. The paper also obtains the Bahadur-Kiefer type representations for these estimators. The class of estimators covered includes the least square, the least absolute deviation, and the Huber(k) estimators