A new wavelet-based estimation of conditional density via block threshold method

Abstract

A new wavelet-based estimator of the conditional density is investigated. The estimator is constructed by combining a special ratio technique and applying a non negative estimator to the density function in the denominator. We used a wavelet shrinkage technique to find an adaptive estimator for this problem. In particular, a block thresholding estimator is proposed, and we prove that it enjoys powerful mean integrated squared error properties over Besov balls. Moreover, it is shown that convergence rates for the mean integrated squared error (MISE) of the adaptive estimator are optimal under some mild assumptions. Finally, a numerical example has been considered to illustrate the performance of the estimator.

Keywords:

• Besov space
• block threshold estimator
• conditional density
• non parametric estimation
• wavelet methods

Acknowledgments

We thank the referee and the Associate Editor for their thorough and useful comments which helped to significantly improve the presentation and results of the article.

Declarations

• There is no funding for this paper

• The authors declare that they have no competing interests.

• This work was carried out in collaboration among all authors. All authors read and approved the final manuscript.

• The data sets used and analyzed during the current study are available from the corresponding author on reasonable request.

• The codes are available from the corresponding author.

Additional information

Funding

Esmaeil Shirazi would like to acknowledge Gonbad Kavous University for the partial support of this research through a Discovery Research Grant. Olivier P. Faugeras acknowledges funding from ANR under grant ANR-17-EURE-0010 (Investissements dAvenir program).