Dynamic Bivariate Normal Copula

TL;DR

This paper extends the understanding of tail dependence in normal copulas by allowing the correlation coefficient to vary with sample size, deriving the asymptotic behavior of maxima, and developing inference methods for the correlation function.

Contribution

It generalizes previous results by deriving the limit distribution of maxima with a variable correlation, and proposes both parametric and nonparametric inference methods for the correlation function.

Findings

Derived the limit of normalized maxima with variable correlation

Developed inference methods for the correlation function

Validated results through simulation and real data analysis

Abstract

Normal copula with a correlation coefficient between $-1$ and $1$ is tail independent and so it severely underestimates extreme probabilities. By letting the correlation coefficient in a normal copula depend on the sample size, H\"usler and Reiss (1989) showed that the tail can become asymptotically dependent. In this paper, we extend this result by deriving the limit of the normalized maximum of $n$ independent observations, where the $i$-th observation follows from a normal copula with its correlation coefficient being either a parametric or a nonparametric function of $i/n$. Furthermore, both parametric and nonparametric inference for this unknown function are studied, which can be employed to test the condition in H\"usler and Reiss (1989). A simulation study and real data analysis are presented too.