Data-driven nonparametric prediction intervals

Standard nonparametric prediction intervals for a single future observation are obtained by taking the interval between two pre-specified order statistics from the initial sample. In this paper, we consider the alternate approach of taking the shortest interval that contains a pre-specified number of the subintervals between the order statistics of the initial sample. We develop a method for determining exact confidence coefficients for such intervals, and we show that these data-driven prediction intervals outperform standard equal-tailed nonparametric prediction intervals. Specifically, they are much shorter than the standard intervals when the underlying distribution is skewed, and they are only slightly longer when the underlying distribution is symmetric. We also obtain the asymptotic approximation that to achieve exact confidence coefficient 1 − α when using the new data-driven prediction intervals with initial sample size n, approximately n ( 1 − α ) + 1.12 n α of the subintervals between the order statistics of the initial sample must be included.