Evaluation of threshold selection methods for adaptive wavelet quantile density estimation in the presence of bias

Abstract

In this paper, the estimation of the quantile density function based on i.i.d biased observations is investigated. The bias function is assumed to be positive and bounded. Of the various smoothing methods for selecting the model parameters, hard and block thresholding methods are proposed and two adaptive estimators based on them are constructed. We evaluate these theoretical performances via the minimax approach over Besov balls. We show that these estimators obtain near-optimal and optimal convergence rates under some mild assumptions. Finally, with a simulation study and application on a real set of data, the performance quality of these estimators will be compared to other wavelet methods.

Keywords:

• Adaptive estimation
• Biased data
• Besov space
• Quantile density function
• Wavelets

Acknowledgments

It is a pleasure to acknowledge helpful comments by the referees and the associated editor which significantly improved the presentation of the paper. The first author would like to acknowledge Gonbad Kavous University for the partial support of this research through a Discovery Research Grant with number 6.00.116.

Disclosure statement

The authors declare that they have no conflict of interest.

Additional information

Funding

The first author was supported by Grant University of Gonbad Kavous with No. 6.252.