Empirical Simultaneous Confidence Regions for Path-Forecasts

A probabilistic assessment about the set of possible trajectories that a random variable may follow over time is summarized by the simultaneous confidence region generated from its forecast generating distribution. However, if the null model is only approximative or altogether unavailable, one cannot derive analytic expressions for this confidence region. Moreover, the high-dimensional nature of the forecast generating distribution in such cases makes non-parametric estimation impractical given commonly available predictive samples. Instead, this paper derives the approximate rectangular confidence regions that control false discovery rate error, which are a function of the predictive sample covariance matrix and the empirical distribution of the Mahalanobis distance of the path-forecast errors. These rectangular regions are simple to construct and appear to work well in a variety of cases explored empirically and by simulation.