Robust estimators and tests for bivariate copulas based on likelihood depth

Estimators and tests based on likelihood depth for one-parametric copulas are given. For the Gaussian and Gumbel copulas, it is shown that the maximum depth estimators are biased. They can be corrected and the new estimators are robust against contamination. For testing, simplicial likelihood depth is considered. Because of the bias of the maximum depth estimator, simplicial likelihood depth is not a degenerated U-statistic so that easily asymptotic α-level tests can be derived for arbitrary hypotheses. Tests are in particular investigated for the one-sided alternatives. Simulation studies for the Gaussian and Gumbel copulas show that the power of the first test is rather good, but the latter one has to be improved, which is also done here. The new tests are robust against contamination.