Semiparametric Multivariate Density Estimation Using Copulas and Shape-Constraints

Maximum likelihood estimation of a log-concave density has certain advantages over other nonparametric approaches, such as kernel density estimation, which requires a bandwidth selection. Furthermore, finding the optimal bandwidth gets more difficult as a dimension increases. On the other hand, the shape-constrained approach is automatic and does not need any tuning parameters. However, for both the kernel and log-concave estimators, the rate of convergence slows down as the dimension d increases. To handle this "curse of dimensionality", we study an intermediate semi-parametric copula approach and we estimate the marginals using the log-concave shape-constrained MLE and use a parametric approach to fit the copula parameters. We prove square root n rate of convergence for the parametric estimator and that the joint density converges at a rate of n^(-2/5) regardless of dimension. This is faster than the conjectured rate of n^(-2/(d+4)) for the multivariate log-concave estimators. We examine the performance of our proposed method via simulation studies and real data example.