A Functional Wavelet-Kernel Approach for Continuous-time Prediction

摘要

We consider the prediction problem of a continuous-time stochastic process onan entire time-interval in terms of its recent past. The approach we adopt isbased on functional kernel nonparametric regression estimation techniques whereobservations are segments of the observed process considered as curves. Thesecurves are assumed to lie within a space of possibly inhomogeneous functions,and the discretized times series dataset consists of a relatively small,compared to the number of segments, number of measurements made at regulartimes. We thus consider only the case where an asymptotically non-increasingnumber of measurements is available for each portion of the times series. Weestimate conditional expectations using appropriate wavelet decompositions ofthe segmented sample paths. A notion of similarity, based on waveletdecompositions, is used in order to calibrate the prediction. Asymptoticproperties when the number of segments grows to infinity are investigated undermild conditions, and a nonparametric resampling procedure is used to generate,in a flexible way, valid asymptotic pointwise confidence intervals for thepredicted trajectories. We illustrate the usefulness of the proposed functionalwavelet-kernel methodology in finite sample situations by means of threereal-life datasets that were collected from different arenas.