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.

翻译：可以认为,最佳预测应该考虑到所有现有数据。 因此, 为了评估预测间隔的性能, 评估预测间隔的性能应该使用有条件的覆盖概率, 并以所有现有观测结果为条件。 我们以线性模型为焦点, 得出名义预测间隔的有条件覆盖概率与通过残留的靴子陷阱获得的有条件覆盖间隔的有条件覆盖概率之间的零星分布。 应用这一结果, 我们发现, 残留的靴子捕捉产生的预测间隔大约有50% 的概率来产生有条件的覆盖不足。 然后我们开发一种新的靴子捕捉算法, 产生一种预测间隔, 从而不同时控制有条件覆盖概率和有条件覆盖不足的可能性。 我们用几种有限模拟来补充无症状结果 。