Asymptotically efficient estimation under local constraint in Wicksell's problem

TL;DR

This paper develops and analyzes informed projection estimators for the Wicksell problem, demonstrating their asymptotic efficiency and normality, and compares them to the isotonic inverse estimator which is shown to be inefficient.

Contribution

It introduces three informed estimators that leverage local constancy, proving their asymptotic efficiency and normality, and clarifies the inefficiency of the isotonic inverse estimator.

Findings

Informed estimators are asymptotically equivalent and normally distributed.

The isotonic inverse estimator is asymptotically inefficient.

Simulation shows informed estimators perform comparably to existing methods.

Abstract

We consider nonparametric estimation of the distribution function $F$ of squared sphere radii in the classical Wicksell problem. Under smoothness conditions on $F$ in a neighborhood of $x$, in \cite{21} it is shown that the Isotonic Inverse Estimator (IIE) is asymptotically efficient and attains rate of convergence $\sqrt{n / \log n}$. If $F$ is constant on an interval containing $x$, the optimal rate of convergence increases to $\sqrt{n}$ and the IIE attains this rate adaptively, i.e.\ without explicitly using the knowledge of local constancy. However, in this case, the asymptotic distribution is not normal. In this paper, we introduce three \textit{informed} projection-type estimators of $F$, which use knowledge on the interval of constancy and show these are all asymptotically equivalent and normal. Furthermore, we establish a local asymptotic minimax lower bound in this setting, proving that the three \textit{informed} estimators are asymptotically efficient and a convolution result showing that the IIE is not efficient. We also derive the asymptotic distribution of the difference of the IIE with the efficient estimators, demonstrating that the IIE is \textit{not} asymptotically equivalent to the \textit{informed} estimators. Through a simulation study, we provide evidence that the performance of the IIE closely resembles that of its competitors.