Conditional Predictive Regions for Stochastic Processes

Studies on pointwise prediction of nonlinear time series have revealed some distinguishing features of nonlinear prediction. Being signiicantly diierent from linear prediction, the prediction accuracy for nonlinear time series depends on the current position in the state space. Furthermore, small perturbation in current position can lead to considerable errors in nonlinear prediction (see Yao and Tong 1994a.) The above features can also be observed in state-dependent interval/region prediction. In this paper, we consider two types of region predictors | the maximum conditional density region and the shortest conditional modal interval, both of them depending on the current position in the state space. When the underlying conditional distribution is skewed or multi-modal, the two proposed predictors outperform the predictive interval constructed from the conditional percentiles. Based on the data f(X t ; Y t); 1 t ng, which are the observations from a strictly stationary process, the conditional distribution function of Y t given X t is estimated using the kernel regression procedure. Further, we estimate the two predictive regions for Y t from X t (t > n) based on the estimated conditional distribution functions. The sensitivity measure deened in the form of the Kullback-Leibler information has been used to monitor the errors in the coverage probabilities of any predictive regions as well as the changes in the width of the conditional modal intervals due to small perturbation in X t. Both simulated and real data sets are used as illustrations.