Robust Prediction Intervals for Time Series Forecasting: A Bootstrap and Bayesian Approach

ABSTRACT This study proposes a new bootstrap strategy, the standardized skewed Student‐ t block multiplier bootstrap (skew‐BMM), designed to better capture heteroskedasticity and heavy‐tailed behavior in time series forecasting with machine learning models. We construct prediction intervals by integrating bootstrap resampling with machine learning forecasting methods and Bayesian approaches, including random forest, support vector regression, XGBoost, and long short‐term memory networks (LSTM), while using the Bayesian structural time series (BSTS) and autoregressive integrated moving average (ARIMA)‐type models as probabilistic benchmark references. To facilitate fair comparisons across models and data‐generating mechanisms, we introduce a scale‐adjusted relative interval score for evaluating interval forecasts. Extensive simulation studies covering nonlinear dynamics, deterministic seasonality, transfer‐function effects, and conditional heteroskedasticity show that the proposed skew‐BMM delivers prediction intervals with improved calibration and competitive sharpness. An empirical application to monthly suicide counts in Japan illustrates the practical relevance of the proposed framework. Overall, the results show that machine learning models combined with the skew‐BMM, particularly LSTM, deliver reliable and accurate forecast distributions in complex and heavy‐tailed environments, while BSTS performs particularly well for probabilistic forecasting.