Bootstrap prediction intervals with asymptotic conditional validity and unconditional guarantees. (14th June 2022)

• Record Type: Journal Article
• Authors: Zhang, Yunyi; Politis, Dimitris N
• Is Part Of: Information and inference. Volume 12:Number 1(2023)
• Page Start: 157
• Page End: 209
• Publication Date: 2022-06-14
• Subjects: prediction -- regression -- bootstrap -- conditional validity; Mathematical models -- Periodicals; 519.605
• DOI: 10.1093/imaiai/iaac017
• Languages: English
• ISSNs: 2049-8764

Abstract

Abstract: It can be argued that optimal prediction should take into account all available data. Therefore, to evaluate a prediction interval's performance one should employ conditional coverage probability, conditioning on all available observations. Focusing on a linear model, we derive the asymptotic distribution of the difference between the conditional coverage probability of a nominal prediction interval and the conditional coverage probability of a prediction interval obtained via a residual-based bootstrap. Applying this result, we show that a prediction interval generated by the residual-based bootstrap has approximately $50\%$ probability to yield conditional under-coverage. We then develop a new bootstrap algorithm that generates a prediction interval that asymptotically controls both the conditional coverage probability as well as the possibility of conditional under-coverage. We complement the asymptotic results with several finite-sample simulations.