It is a standard assumption in the density deconvolution problem that the characteristic function of the measurement error distribution is non-zero on the real line. While this condition is assumed in the majority of existing works on the topic, there are many problem instances of interest where it is violated. In this paper we focus on non-standard settings where the characteristic function of the measurement errors has zeros, and study how zeros multiplicity affects the estimation accuracy. For a prototypical problem of this type we demonstrate that the best achievable estimation accuracy is determined by the multiplicity of zeros, the rate of decay of the error characteristic function, as well as by the smoothness and the tail behavior of the estimated density. We derive lower bounds on the minimax risk and develop optimal in the minimax sense estimators. In addition, we consider the problem of adaptive estimation and propose a data-driven estimator that automatically adapts to unknown smoothness and tail behavior of the density to be estimated.
Bibliographical noteFunding Information:
∗The research was supported by the Israel Science Foundation (ISF) grant.
© 2021, Institute of Mathematical Statistics. All rights reserved.
- Characteristic function
- Density deconvolution
- Laplace transform
- Minimax risk
- Non-standard measurement error
- Zero multiplicity
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty