  The statistical theory of extremes is extended to observations that are
non-stationary and not independent. The non-stationarity over time and space is
controlled via the scedasis (tail scale) in the marginal distributions. Spatial
dependence stems from multivariate extreme value theory. We establish
asymptotic theory for both the weighted sequential tail empirical process and
the weighted tail quantile process based on all observations, taken over time
and space. The results yield two statistical tests for homoscedasticity in the
tail, one in space and one in time. Further, we show that the common extreme
value index can be estimated via a pseudo-maximum likelihood procedure based on
pooling all (non-stationary and dependent) observations. Our leading example
and application is rainfall in Northern Germany.
