Wavelet block thresholding for copula density estimation under biased sampling

Abstract

In this paper, we consider the non-parametric estimation of the copula density under biased data. The contributions are both theoretical and practical. In the first part, we propose and develop a new wavelet-based methodology for this problem. In particular, a BlockShrink estimator is constructed, and we prove that it enjoys powerful mean integrated squared error properties over Besov balls. The second part is devoted to the applied aspect: we compare the performance of the wavelet-based estimator with that of a recently introduced kernel-based estimator through a detailed simulation study.

Keywords:

• Biased data
• copula
• non-parametric estimation
• wavelets
• weighted distributions

Acknowledgments

The authors thank the referees and the associated editor for insightful comments that helped them to improve the article significantly. This work was carried out in collaboration among all authors. All authors read and approved the final manuscript. The data sets used and analysed during the current study are available from the corresponding author on reasonable request. The codes are available from the corresponding author.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Esmaeil Shirazi would like to acknowledge Gonbad Kavous Universiy for the partial support of this research through a Discovery Research Grant.