Wavelet estimations for heteroscedastic super smooth errors

Abstract

The observed data are usually contaminated by various errors in practical applications. This paper deals with the density deconvolution problems with super smooth errors under heteroscedastic situation. We provide a new wavelet estimator and investigate its upper bound of L_p risk over Besov ball . Then the lower bound is given which shows that the defined estimator attains the optimal convergence rate. It turns out that Theorem 3.1 extends some existing theorems of Pensky and Vidakovic, Fan and Koo, as well as Li and Liu in some sense. However, since the convergence is just logarithmic rate which is not satisfactory for us. Motivated by the works of Pensky and Vidakovic, Li and Liu, we study the wavelet estimator over super smooth space to improve the convergence rate.

Keywords:

• Besov space
• super smooth space
• heteroscedastic error
• wavelet estimator
• convergence rate

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G07
• 62G20
• 42C40

Additional information

Funding

This paper is supported by the National Natural Science Foundation of China (No. 11771030). The authors would like to thank referees and an editor for their good suggestions and comments.