Nonparametric bivariate copula estimation based on shape-restricted support vector regression

Copula has become a standard tool in describing dependent relations between random variables. This paper proposes a nonparametric bivariate copula estimation method based on shape-restricted -support vector regression ( -SVR). This method explicitly supplements the classical -SVR with constraints related to three shape restrictions: grounded, marginal and 2-increasing, which are the necessary and sufficient conditions for a bivariate function to be a copula. This nonparametric method can be reformulated to a convex quadratic programming, which is computationally tractable. Experiments on both five artificial data sets and three international stock indexes clearly showed that it could achieve significantly better performance than common parametric models and kernel smoother. 2012 Elsevier B.V. All rights reserved.