Sampling properties and empirical estimates of extreme events

Ed Mackay,Philip Jonathan

doi:10.1016/j.oceaneng.2021.109791

Abstract

The statistical characteristics of the largest observations in a sample are highly uncertain. In this work we consider the problem of how to define empirical estimates of exceedance probabilities and return periods associated with an ordered sample of observations. Understanding the sampling properties of these quantities is important for assessing the fit of a statistical model and also for placing confidence bounds on estimates of extreme events from Monte Carlo simulations. The empirical distribution function (EDF) is often defined as the expected non-exceedance probability (NEP) associated with sample order statistics. Yet, due to the non-linearity of the relations between return periods, quantiles and NEP, the return period (or quantile) associated with the expected NEP is not equal to the expected return period (or quantile), leading to ambiguity. However, the sampling distributions of exceedance probabilities, return periods and quantiles are, in fact, linked by a simple relation. From this relation, it follows that defining the EDF in terms of the median NEP of the order statistics gives a consistent framework for defining empirical estimates of all three quantities. We demonstrate that the median value of the return period of the largest observation is 44% larger than the return period calculated using the common definition of the EDF in terms of the expected NEP of the order statistics. We also derive some new results about the size of the confidence intervals for exceedance probabilities and return periods.

Highlights

Estimating the frequency of occurrence of extreme events is an important topic in offshore and coastal engineering
We argue that a common framework can be used for defining the empirical estimates of either exceedance probabilities, return periods or quantiles, where the empirical distribution function (EDF) is defined as pk = median(Pk), rather than pk = E(Pk)
We show that the sampling distributions of the exceedance probabilities, return periods and quantiles are all linked by a simple relation, that makes the use of the median value appropriate for all cases

Summary

Introduction

Estimating the frequency of occurrence of extreme events is an important topic in offshore and coastal engineering. The fit of the model is usually assessed using plots of the observations together with various quantities derived from the empirical distribution function (EDF), such as exceedance probabilities, return periods or quantiles. We show that the sampling distributions of the exceedance probabilities, return periods and quantiles are all linked by a simple relation, that makes the use of the median value appropriate for all cases This relation is used to derive some results about the confidence intervals associated with extreme observations. If we define empirical estimates of probabilities, return periods and quantiles in terms of the medians of the sampling distribution, this gives a consistent framework that can be used in all types of model diagnostic plots, as discussed further .

Diagnostic plots for extreme value models

Confidence intervals

Relation between CI for probabilities and CI for quantiles

Findings

Conclusions