Tolerance intervals for autoregressive models, with an application to hospital waiting lists

Long waiting lists are a symbol of inefficiencies of hospital services. The dynamics of waiting lists are complex, especially when trying to understand how the lists grow due to the demand of a particular treatment relative to a hospital's capacity. Understanding the uncertainty of forecasting growth/decline of waiting lists could help hospital managers with capacity planning. We address this uncertainty through the use of statistical tolerance intervals, which are intervals that contain a specified proportion of the sampled population at a given confidence level. Tolerance intervals are available for numerous settings, however, the approaches for autoregressive models are far more limited. This article fills that gap and establishes tolerance intervals for general AR(p) models, which may also have a mean or trend component present. A rigorous development of tolerance intervals in this setting is presented. Extensive simulation studies identify that good coverage properties are achieved when the AR process is stationary and the parameters of the AR model are well within the stationarity constraints. Otherwise, a bootstrap-based correction can be applied to improve the coverage probabilities. Finally, the method is applied to the monthly number of patients on hospital waiting lists in England.