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The Copula is widely used to describe the relationship between the marginaldistribution and joint distribution of random variables. The estimation ofhigh-dimensional Copula is difficult, and most existing solutions rely eitheron simplified assumptions or on complicating recursive decompositions.Therefore, people still hope to obtain a generic Copula estimation method withboth universality and simplicity. To reach this goal, a novel neuralnetwork-based method (named Neural Copula) is proposed in this paper. In thismethod, a hierarchical unsupervised neural network is constructed to estimatethe marginal distribution function and the Copula function by solvingdifferential equations. In the training program, various constraints areimposed on both the neural network and its derivatives. The Copula estimated bythe proposed method is smooth and has an analytic expression. The effectivenessof the proposed method is evaluated on both real-world datasets and complexnumerical simulations. Experimental results show that Neural Copula's fittingquality for complex distributions is much better than classical methods. Therelevant code for the experiments is available on GitHub. (We encourage thereader to run the program for a better understanding of the proposed method).

Neural Copula: A unified framework for estimating generic high-dimensional Copula functions