  We investigate two models for the following setup: We consider a stochastic
process X \in C[0,1] whose distribution belongs to a parametric family indexed
by \vartheta \in {\Theta} \subset R. In case \vartheta = 0, X is a generalized
Pareto process. Based on n independent copies X(1),...,X(n) of X, we establish
local asymptotic normality (LAN) of the point process of exceedances among
X(1),...,X(n) above an increasing threshold line in each model.
  The corresponding central sequences provide asymptotically optimal sequences
of tests for testing H0 : \vartheta = 0 against a sequence of alternatives Hn :
\vartheta = \varthetan converging to zero as n increases. In one model, with an
underlying exponential family, the central sequence is provided by the number
of exceedances only, whereas in the other one the exceedances themselves
contribute, too. However it turns out that, in both cases, the test statistics
also depend on some additional and usually unknown model parameters.
  We, therefore, consider an omnibus test statistic sequence as well and
compute its asymptotic relative efficiency with respect to the optimal test
sequence.
