Copula density estimation by total variation penalized likelihood with linear equality constraints

• Published in: Computational statistics & data analysis Vol. 56; no. 2; pp. 384 - 398
• Main Authors: Qu, Leming, Yin, Wotao
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.02.2012; Elsevier
• Series: Computational Statistics & Data Analysis
• Subjects: Algorithms; Augmented Lagrangian method; Computational efficiency; Computer simulation; Copula density estimation; Copula density estimation Total variation Maximum penalized likelihood estimation Augmented Lagrangian method; Density; Exact sciences and technology; General topics; Mathematics; Matlab; Maximum penalized likelihood estimation; Multivariate analysis; Nonparametric inference; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in probability and statistics; Probability and statistics; Probability density functions; Sciences and techniques of general use; Statistics; Television; Total variation; Total variation; Augmented Lagrangian method; Copula density estimation; Maximum penalized likelihood estimation; Rank statistic; Density estimation; Optimization method; Non parametric estimation; Cross validation; Random vector; Multivariate analysis; Penalty method; Parametric method; Linear constraint; Marginal distribution; Operator splitting; Probability density function; Ill posed problem; Lagrangian method; Symmetry; Copulas; Discriminant analysis; Data analysis; Lagrangian; Numerical linear algebra; Statistical estimation; Algorithm; Constrained optimization; Regularization method; Selection problem; Numerical analysis; Statistical computation; Simulation; Joint probability; Equality constraint; Maximum likelihood; Regularization
• ISSN: 0167-9473; 1872-7352
• DOI: 10.1016/j.csda.2011.07.016

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 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 the log-barrier method for the 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.