Extremal Characteristics of Conditional Models

TL;DR

This paper develops methods to analyze the extremal behavior of bivariate conditional models, providing tools to approximate tail distributions and asymptotic independence coefficients, with applications to wave data and theoretical insights.

Contribution

It introduces novel tools for quantifying extremal characteristics of conditional models, including closed-form approximations and a new parameter space condition.

Findings

Tools for approximating tail distributions of Y

Method to estimate asymptotic independence coefficient η

Conditional extremes model cannot capture η<1

Abstract

Conditionally specified models are often used to describe complex multivariate data. Such models assume implicit structures on the extremes. So far, no methodology exists for calculating extremal characteristics of conditional models since the copula and marginals are not expressed in closed forms. We consider bivariate conditional models that specify the distribution of $X$ and the distribution of $Y$ conditional on $X$. We provide tools to quantify implicit assumptions on the extremes of this class of models. In particular, these tools allow us to approximate the distribution of the tail of $Y$ and the coefficient of asymptotic independence $\eta$ in closed forms. We apply these methods to a widely used conditional model for wave height and wave period. Moreover, we introduce a new condition on the parameter space for the conditional extremes model of Heffernan and Tawn (2004), and prove that the conditional extremes model does not capture $\eta$, when $\eta<1$.