Selection of Mixed Copula Model via Penalized Likelihood

A fundamental issue of applying a copula method in applications is how to choose an appropriate copula function for a given problem. In this article we address this issue by proposing a new copula selection approach via penalized likelihood plus a shrinkage operator. The proposed method selects an appropriate copula function and estimates the related parameters simultaneously. We establish the asymptotic properties of the proposed penalized likelihood estimator, including the rate of convergence and asymptotic normality and abnormality. Particularly, when the true coefficient parameters may be on the boundary of the parameter space and the dependence parameters are in an unidentified subset of the parameter space, we show that the limiting distribution for boundary parameter estimator is half-normal and the penalized likelihood estimator for unidentified parameter converges to an arbitrary value. Finally, Monte Carlo simulation studies are carried out to illustrate the finite sample performance of the proposed approach and the proposed method is used to investigate the correlation structure and comovement of financial stock markets.