Reduced-bias estimation of the residual dependence index with unnamed marginals

TL;DR

This work tackles the challenge of estimating the residual tail index $η$ for bivariate extremes under asymptotic independence. It develops a gradient-based, reduced-bias framework that unifies Pareto and shifted unit-Fréchet marginals via a Hall-Welsh-type marginal model, and proves asymptotic normality through a bespoke tail empirical process. A reduced-bias variant leverages a second-order Hall-Welsh expansion to further suppress bias, while preserving variance, and the approach includes a stable threshold-selection mechanism. Through extensive simulations and an application to monsoon rainfall in Ghana, the method demonstrates improved finite-sample performance and provides a practical, threshold-free inference tool for detecting residual dependence in extreme events.

Abstract

This paper addresses important weaknesses in current methodology for the estimation of multivariate extreme event distributions. The estimation of the residual dependence index $η\in (0,1]$ is notoriously problematic. We introduce a flexible class of reduced-bias estimators for this parameter, designed to ameliorate the usual problems of threshold selection through a unified approach to familiar marginal standardisations. We derive the asymptotic properties of the proposed class of gradient estimators for $η$. Their efficiency stems from a hitherto neglected exponentially decaying term in the characterisation of the asymptotic independence based on the theory of regular variation. Simulation studies to demonstrate the finite-sample efficacy of the new gradient estimation across a wealth of bivariate distributions belonging to some max-domain of attraction that enjoy the asymptotic independence property. Our leading application illustrates how asymptotic independence can be discerned from monsoon-related rainfall occurrences at different locations in Ghana. The considerations involved in extending this framework to the estimation of the extreme value index attached to univariate domains of attraction associated with heavy-tailed distributions are briefly discussed.

Figures (8)

• Figure 1: Case of $b=-a$: asymptotic variance of $\hat{\eta}^{S}_{a,b}$ for true $\eta \in (0, 0.8]$, with $a \in [-0.4,\, 0.4]$.
• Figure 2: Map of southern Ghana highlighting the three selected stations.
• Figure 3: Scatter plots of positive daily rainfall transformed to the unit uniform scale for the two pairs of stations: (a) (ASU, MAM); (b) (ASU, BRI).
• Figure 4: Well-separated stations (ASU, MAM): plot (a) concerns reduced-bias estimation of the residual dependence index $\eta \in (0,1]$. The blue thick line gives the sample path for $q=0.5$, whereas the red solid line corresponds to the $q=1.5$. Shaded areas represent their respective $95\%$ confidence bands. The yellow line for $q=1$ corresponds to the Hill estimator; plot (b) is the analogous plot for the plain estimator, with no bias reduction employed.
• Figure 5: Nearby stations (ASU, BRI): plot (a) concerns reduced-bias estimation of the residual dependence index $\eta \in (0,1]$. The blue thick line gives the sample path for $q=0.5$, whereas the red solid line corresponds to the $q=1.5$. Shaded areas represent their respective $95\%$ confidence bands. The yellow line for $q=1$ corresponds to the Hill estimator; plot (b) is the analogous plot for the plain gradient estimator, with no bias reduction employed.

Theorems & Definitions (15)

• Example 1
• Theorem 2.1
• Remark 1
• Remark 2
• Theorem 2.2
• Remark 3
• Theorem 3.1
• Remark 4
• Theorem 5.1
• Theorem 5.2