Transformation-based nonparametric estimation of multivariate densities

We present a probability-integral-transformation-based estimator of multivariate densities. Given a sample of random vectors, we first transform the data into their corresponding marginal distributions. The marginal densities and the joint density of the transformed data are estimated nonparametrically. The joint density of the original data is constructed as the product of the density of the transformed data and marginal densities, which coincides with the copula representation of multivariate densities. We show that the Kullback–Leibler Information Criterion (KLIC) between the true density and its estimate can be decomposed into the KLIC of the marginal densities and that of the copula density. We derive the convergence rate of the proposed estimator in terms of the KLIC and propose a supervised hierarchical procedure of model selection. Monte Carlo simulations demonstrate the good performance of the estimator. An empirical example on the US and UK stock markets is presented. The estimated conditional copula density provides useful insight into the joint movements of the US and UK markets under extreme Asian markets.