  We develop methods, based on extreme value theory, for analysing observations
in the tails of longitudinal data, i.e., a data set consisting of a large
number of short time series, which are typically irregularly and
non-simultaneously sampled, yet have some commonality in the structure of each
series and exhibit independence between time series. Extreme value theory has
not been considered previously for the unique features of longitudinal data.
Across time series the data are assumed to follow a common generalised Pareto
distribution, above a high threshold. To account for temporal dependence of
such data we require a model to describe (i) the variation between the
different time series properties, (ii) the changes in distribution over time,
and (iii) the temporal dependence within each series. Our methodology has the
flexibility to capture both asymptotic dependence and asymptotic independence,
with this characteristic determined by the data. Bayesian inference is used
given the need for inference of parameters that are unique to each time series.
Our novel methodology is illustrated through the analysis of data from elite
swimmers in the men's 100m breaststroke. Unlike previous analyses of
personal-best data in this event, we are able to make inference about the
careers of individual swimmers - such as the probability an individual will
break the world record or swim the fastest time next year.
