Testing for a generalized Pareto process

TL;DR

This paper develops statistical tests for generalized Pareto processes by analyzing exceedances in stochastic processes, establishing asymptotic properties, and comparing the efficiency of different test procedures.

Contribution

It introduces asymptotically optimal tests for generalized Pareto processes based on exceedance data and evaluates their efficiency relative to omnibus tests.

Findings

Established local asymptotic normality for exceedance-based models.

Derived asymptotically optimal test sequences for hypothesis testing.

Compared efficiency of different test procedures, including omnibus tests.

Abstract

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.