Exceedance probabilities using Nonparametric Predictive Inference

Abstract

Some statistical methods for extreme value analysis assume that the maximum observed value represents the endpoint of the support. However, in cases involving right-censored observations, it is often unclear whether the true value of a censored observation exceeds the largest observed value. This paper is inspired by the Supercentenarian dataset, which contains the ages at death of individuals who lived beyond 110 years, with right-censored data for those still alive at the time of data collection. This study employs Nonparametric Predictive Inference (NPI), a method that provides probability statements for a range of events of interest. NPI is a frequentist method that relies on minimal assumptions, focusing explicitly on future observations. It uses imprecise probabilities based on Hill’s assumption $A_{\left(n\right)}$ to quantify uncertainty. In this paper, we derive the probability that the true lifetime of at least one right-censored observation – or one of the future observations – exceeds the largest observed value. Furthermore, we extend this analysis to the exceedance of multiple largest observations, provided that they exceed the largest censored observation. We also investigate the time between any two of these largest observations, deriving the lower and upper probabilities for the exceedance of this time. We then demonstrate the proposed method using the Supercentenarian dataset, where the analysis is performed separately for men and women. We show how this approach can help to assess the likelihood of future extreme observations and provide insights into the validity of assuming the largest observed value as the endpoint of support. This work highlights the strengths of NPI in handling right-censored data and its application to real-world datasets.