Parametric likelihood inference for record breaking problems

SUMMARY: In this paper we consider the analysis of record breaking data sets, where only observations that exceed, or only those that fall below, the current extreme value are recorded. Example application areas include industrial stress testing, meteorological analysis, sporting and athletic events, and oil and mining surveys. A closely related area is that of threshold modelling, where the observations are those that cross a certain threshold value. The inherent missing data structure present in these problems leads to likelihood functions that contain possibly high-dimensional integrals, rendering traditional maximum likelihood methods difficult or not feasible. Fortunately, we may obtain arbitrarily accurate approximations to the likelihood function by iteratively applying Monte Carlo integration methods (Geyer & Thompson, 1992). Subiteration using the Gibbs sampler may help to evaluate any multivariate integrals encountered during this process. This approach can handle far more sophisticated parametric models than have been used previously in record breaking and threshold data contexts. In particular, the methodology allows for observations that are dependent and subject to mean shifts over time. We present a numerical example involving records in Olympic high jump competition, where besides estimation we also address related issues in model selection and prediction.