Model-Free Inference for Markov Processes

Abstract

Given time series data $Y _{1},\ldots,Y _{n}$, the problem of optimal prediction of Y _n+1 has been well-studied. The same is not true, however, as regards the problem of constructing a prediction interval with prespecified coverage probability for Y _n+1, i.e., turning the point predictor into an interval predictor. In Pan and Politis (J Stat Plann Inference, 2015, to appear), the case where the time series {Y _t } obeys an autoregressive model was addressed in detail with the autoregression allowed to be linear, nonlinear or nonparametric. In the paper at hand, we expand the scope by assuming only that {Y _t } is a Markov process of order p ≥ 1 without insisting that any specific autoregressive equation is satisfied. Several different approaches and methods are considered, namely both Forward and Backward approaches to prediction intervals as combined with three resampling methods: the bootstrap based on estimated transition densities, the Local Bootstrap for Markov processes, and the novel Model-Free bootstrap. In simulations, prediction intervals obtained from different methods are compared in terms of their coverage level and length of interval.