Sobolev Convergence of Empirical Bernstein Copulas

In this work, we prove that Bernstein estimator always converges to the true copula under Sobolev distances. The rate of convergences is provided in case the true copula has bounded second order derivatives. Simulation study has also been done for Clayton copulas. We then use this estimator to estimate measures of complete dependence for weather data. The result suggests a nonlinear relationship between the dust density in Chiang Mai, Thailand and the temperature and the humidity level.