Penalized Exponential Series Estimation of Copula Densities

The exponential series density estimator is advantageous to the copula density estimation as it is strictly positive, explicitly defined on a bounded support, and largely mitigates the boundary bias problem. However, the selection of basis functions is challenging and can cause numerical difficulties, especially for high dimensional density estimations. To avoid the issues associated with basis function selection, we adopt the strategy of regularization by employing a relatively large basis and penalizing the roughness of the resulting model, which leads to a penalized maximum likelihood estimator. To further reduce the computational cost, we propose an approximate likelihood cross validation method for the selection of smoothing parameter. Our extensive Monte Carlo simulations demonstrate the effectiveness of the proposed estimator for copula density estimations. ∗Department of Agricultural Economics, Texas A&M University. Email: [email protected]. I gratefully acknowledge the Supercomputing Facility of the Texas A&M University where all computations in the study were performed.