Model misspecification in peaks over threshold analysis

TL;DR

This paper investigates the impact of model misspecification in peaks over threshold analysis, proposing new estimators, tests for model validity, and illustrating their application with real climate data.

Contribution

It extends estimators of the extremal index, proves their statistical properties, and develops tests for model misspecification and parameter choice in peaks over threshold analysis.

Findings

New estimators for the extremal index with proven consistency and asymptotic normality.

Development of misspecification tests for model validity and parameter selection.

Successful application to climate data demonstrating estimation and validation methods.

Abstract

Classical peaks over threshold analysis is widely used for statistical modeling of sample extremes, and can be supplemented by a model for the sizes of clusters of exceedances. Under mild conditions a compound Poisson process model allows the estimation of the marginal distribution of threshold exceedances and of the mean cluster size, but requires the choice of a threshold and of a run parameter, $K$, that determines how exceedances are declustered. We extend a class of estimators of the reciprocal mean cluster size, known as the extremal index, establish consistency and asymptotic normality, and use the compound Poisson process to derive misspecification tests of model validity and of the choice of run parameter and threshold. Simulated examples and real data on temperatures and rainfall illustrate the ideas, both for estimating the extremal index in nonstandard situations and for assessing the validity of extremal models.