Transformation-Kernel Estimation of Copula Densities

ABSTRACT

The standard kernel estimator of copula densities suffers from boundary biases and inconsistency due to unbounded densities. Transforming the domain of estimation into an unbounded one remedies both problems, but also introduces an unbounded multiplier that may produce erratic boundary behaviors in the final density estimate. We propose an improved transformation-kernel estimator that employs a smooth tapering device to counter the undesirable influence of the multiplier. We establish the theoretical properties of the new estimator and its automatic higher-order improvement under Gaussian copulas. We present two practical methods of smoothing parameter selection. Extensive Monte Carlo simulations demonstrate the competence of the proposed estimator in terms of global and tail performance. Two real-world examples are provided. Supplementary materials for this article are available online.

KEY WORDS:

• Boundary bias
• Copula density
• Kernel density estimation
• Transformation.

Additional information

Funding

The authors thank the associate editor and two anonymous referees for their valuable comments and suggestions. The first author acknowledges financial support from the National Natural Science Foundation of China (No. 71703108; No. 71602128).