Shrinkage for Gaussian and t Copulas in Ultra-High Dimensions

Copulas are a convenient framework to synthesize joint distributions, particularly in higher dimensions. Currently, copula-based high dimensional settings are used for as many as a few hundred variables and require large data samples for estimation to be precise. In this paper, we employ shrinkage techniques for large covariance matrices in the problem of estimation of Gaussian and t copulas whose dimensionality goes well beyond that typical in the literature. Specifically, we use the covariance matrix shrinkage of Ledoit and Wolf to estimate large matrix parameters of Gaussian and t copulas for up to thousands of variables, using up to 20 times lower sample sizes. The simulation study shows that the shrinkage estimation significantly outperforms traditional estimators, both in low and especially high dimensions. We also apply this approach to the problem of allocation of large portfolios.