Paper detail

Online Tracking of a Predictable Drifting Parameter of a Time Series

We propose an online algorithm for tracking a multidimensional time-varying parameter of a time series, which is also allowed to be a predictable process with respect to the underlying time series. The algorithm is driven by a gain function. Under assumptions on the gain, we derive uniform non-asymptotic error bounds on the tracking algorithm in terms of chosen step size for the algorithm and the variation of the parameter of interest. We also outline how appropriate gain functions can be constructed. We give several examples of different variational setups for the parameter process where our result can be applied. The proposed approach covers many frameworks and models (including the classical Robbins-Monro and Kiefer-Wolfowitz procedures) where stochastic approximation algorithms comprise the main inference tool for the data analysis. We treat in some detail a couple of specific models.