In this paper we consider the analysis of record breaking data sets, w
here 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 athl
etic events, and oil and mining surveys. A closely related area is tha
t 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 arbitrar
ily accurate approximations to the likelihood function by iteratively
applying Monte Carlo integration methods (Geyer & Thompson, 1992). Sub
iteration using the Gibbs sampler may help to evaluate any multivariat
e integrals encountered during this process. This approach can handle
far more sophisticated parametric models than have been used previousl
y in record breaking and threshold data contexts. In particular, the m
ethodology allows for observations that are dependent and subject to m
ean shifts over time. We present a numerical example involving records
in Olympic high jump competition, where besides estimation we also ad
dress related issues in model selection and prediction.