A copula density is the joint probability density function (PDF) of a random vector with uniform marginals. An approach to bivariate copula density estimation is introduced that is based on a maximum penalized likelihood estimation (MPLE) with a total variation (TV) penalty term. The marginal unity and symmetry constraints for copula density are enforced by linear equality constraints. The TV-MPLE subject to linear equality constraints is solved by an augmented Lagrangian and operator-splitting algorithm. It offers an order of magnitude improvement in computational efficiency over another TV-MPLE method without constraints solved by log-barrier method for second order cone program. A data-driven selection of the regularization parameter is through K-fold cross-validation (CV). Simulation and real data application show the effectiveness of the proposed approach. The MATLAB code implementing the methodology is available online.
Qu, Leming and Yin, Wotao. (2011). "Copula Density Estimation by Total Variation Penalized Likelihood with Linear Equality Constraints". Computational Statistics & Data Analysis, 56(2), 384-398. http://dx.doi.org/10.1016/j.csda.2011.07.016