Outer and inner prediction intervals for order statistics intervals based on current records

Abstract

This paper considers the largest and smallest observations at the times when a new record of either kind (upper or lower) occurs. These are called the upper and lower current records and are denoted by ${R^l_m}$ and ${R^s_m}$, respectively. The interval ${(R^s_m,R^l_m)}$ is then referred to as the record coverage. The prediction problem in the two-sample case is then discussed and, specifically, the exact outer and inner prediction intervals are derived for order statistics intervals from an independent future Y-sample based on the m-th record coverage from the X-sequence when the underlying distribution of the two samples are the same. The coverage probabilities of these intervals are exact and do not depend on the underlying distribution. Distribution-free prediction intervals as well as upper and lower prediction limits for spacings from a future Y-sample are obtained in terms of the record range from the X-sequence.