Conditional empirical, quantile and difference processes for a large class of time series with applications

Abstract

Abstract The speed of a.s. convergence of conditional empirical distribution functions, quantile functions and difference processes is studied for a wide class of stationary time series. The difference process leads to a Bahadur-Kiefer representation of the conditional quantiles. Applications like autoregression function estimation, estimation of the hazard rate under random censoring, and robust conditional location functionals are also included. When specializing to the i.i.d. case our rates are very close to the best rates known in that special situation. Thus, some recent results on conditional processes with applications are extended to models with dependent data. Rather than modeling this dependence structure by the usual mixing conditions which are hard to precisely understand intuitively since they may fail to hold in very decent cases, we use the notion of mn-decomposability. Time series included are linear (with generalizes ARMA) processes and, more generally, processes with a Volterra expansion ofany finite 1order. Also bilinear processes are in this class.