Density Deconvolution with Non-Standard Error Distributions: Rates of Convergence and Adaptive Estimation

TL;DR

This paper investigates density deconvolution when measurement error characteristic functions have zeros, analyzing how zeros' multiplicity impacts estimation accuracy and proposing adaptive estimators for such non-standard cases.

Contribution

It introduces a theoretical framework for density deconvolution with zero-containing error characteristic functions, deriving minimax bounds and developing adaptive estimators.

Findings

Zeros' multiplicity affects estimation rates

Derived minimax lower bounds for non-standard errors

Proposed adaptive estimator for unknown smoothness

Abstract

It is a typical 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.