A Novel Nonparametric Estimation for Conditional Copula Functions Based on Bayes Theorem

Conditional copula which measures the conditional dependence among variables, possesses a special position in copula field. In this article, based on Bayes theorem, we derive three kinds of conditional copula functions as the product of the corresponding conditional copula density functions and the corresponding unconditional copula functions (or the cumulative distribution functions). Then, a novel nonparametric method for estimating these conditional copula functions is proposed by the classification of the Monte Carlo Simulation (MCS) samples and by the kernel density estimation. In contrast to other estimation methods for conditional copula functions, the proposed method needs only a set of samples without any parameter or distribution assumption, or other complicated operators (such as estimation of the weights, integral operator, etc.). Therefore, the proposed nonparametric method reduces the computational complexity and possesses more universality for estimating the conditional copula functions. A 2-dimensional normal copula function, a numerical example, a structural system reliability analysis considering the common cause failure and an astrophysics model based on real data are employed to validate the effectiveness of the proposed method. Results show that the proposed nonparametric method is accurate and practical well.