Extreme Value Estimation for Discretely Sampled Continuous Processes

TL;DR

This paper investigates how discretely sampled data affects extreme value estimators for continuous stochastic processes, establishing conditions under which interpolated estimators remain asymptotically valid.

Contribution

It provides theoretical conditions on sampling schemes ensuring interpolated estimators perform as well as fully observed data in extreme value analysis.

Findings

Interpolated estimators can be asymptotically equivalent to continuous observations under certain sampling conditions.

Conditions on the fineness of sampling are derived to guarantee estimator consistency.

The results guide practical data collection for environmental extreme value modeling.

Abstract

In environmental applications of extreme value statistics, the underlying stochastic process is often modeled either as a max-stable process in continuous time/space or as a process in the domain of attraction of such a max-stable process. In practice, however, the processes are typically only observed at discrete points and one has to resort to interpolation to fill in the gaps. We discuss the influence of such an interpolation on estimators of marginal parameters as well as estimators of the exponent measure. In particular, natural conditions on the fineness of the observational scheme are developed which ensure that asymptotically the interpolated estimators behave in the same way as the estimators which use fully observed continuous processes.