Rank-based inference for bivariate extreme-value copulas

TL;DR

This paper introduces rank-based estimators for the Pickands dependence function in bivariate extreme-value copulas, applicable when marginal distributions are unknown, and analyzes their theoretical properties and performance.

Contribution

It develops and analyzes rank-based estimators for the dependence function when margins are unknown, extending existing methods that assume known margins.

Findings

Establishes consistency and asymptotic normality of the proposed estimators.

Demonstrates comparable finite-sample performance to known-margin estimators.

Provides explicit formulas for asymptotic variances and their estimates.

Abstract

Consider a continuous random pair $(X,Y)$ whose dependence is characterized by an extreme-value copula with Pickands dependence function $A$. When the marginal distributions of $X$ and $Y$ are known, several consistent estimators of $A$ are available. Most of them are variants of the estimators due to Pickands [Bull. Inst. Internat. Statist. 49 (1981) 859--878] and Cap\'{e}ra\`{a}, Foug\`{e}res and Genest [Biometrika 84 (1997) 567--577]. In this paper, rank-based versions of these estimators are proposed for the more common case where the margins of $X$ and $Y$ are unknown. Results on the limit behavior of a class of weighted bivariate empirical processes are used to show the consistency and asymptotic normality of these rank-based estimators. Their finite- and large-sample performance is then compared to that of their known-margin analogues, as well as with endpoint-corrected versions thereof. Explicit formulas and consistent estimates for their asymptotic variances are also given.