Dynamic Bivariate Normal Copula

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, Husler 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 Husler and Reiss (1989). A simulation study and real data analysis are presented too.

Rates of convergence of extremes from skew normal samples

For a skew normal random sequence, convergence rates of the distribution of its partial maximum to the Gumbel extreme value distribution are derived. The asymptotic expansion of the distribution of the normalized maximum is given under an optimal choice of norming constants. We find that the optimal convergence rate of the normalized maximum to the Gumbel extreme value distribution is proportional to $1/log n$.