The asymptotic behaviors for autoregression quantile estimates

This article is concerned with the asymptotic theory of estimates of unknown parameters in autoregressive quantile processes. We assume random errors form a strictly stationary ϕ-mixing sequences. In view of the approach of argmins and blocking argument, we prove the parameter estimators satisfy the functional moderate deviation principle (MDP). Further, we give the law of the iterated logarithm under some standard conditions. Based on the contraction principle, the moderate deviation principles of L-estimators on the autoregression quantile (ARQ) and autoregression rank scores (ARRS’s) are also discussed. This method can be extended to a fair range of different statistical estimation problems.