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Asymptotically efficient estimation under local constraint in Wicksell's problem

Francesco Gili, Geurt Jongbloed, Aad van der Vaart

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.

Asymptotically efficient estimation under local constraint in Wicksell's problem

Abstract

We consider nonparametric estimation of the distribution function of squared sphere radii in the classical Wicksell problem. Under smoothness conditions on in a neighborhood of , in \cite{21} it is shown that the Isotonic Inverse Estimator (IIE) is asymptotically efficient and attains rate of convergence . If is constant on an interval containing , the optimal rate of convergence increases to 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 , 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.

Paper Structure

This paper contains 8 sections, 11 theorems, 131 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Construction of the estimators
  3. Asymptotic distributions of the estimators
  4. Lower bound for the local asymptotic minimax risk
  5. Simulation study
  6. Conclusion
  7. Proofs.
  8. Efficiency Theory

Key Result

Proposition 1

Let $U_n$ be as in eq: U_n, $\underaccent{\bar{}}{x}<\bar{x}$ with $[\underaccent{\bar{}}{x},\bar{x}) \in \mathscr{P}$ and $x \in [0,\infty)$. The solutions of the minimizations eq: isotonic estimator as proj-eq: proj naive smaller space are: where for $x \in [x_i,x_{i+1}) \in \mathscr{P}$: where:

Figures (5)

  • Figure 1: Underlying cdf $F$ and in light blue a histogram of a sample of size $1000$ from $g$.
  • Figure 2: All estimators and the true function $V$, which constant on $[2,3]$, for a sample of size $n=300$.
  • Figure 3: Scatter and Tukey mean-difference plots for $n=1000$ and $300$ repetitions of $\hat{V}_n$ and $V^{(\underline{x},\overline{x})}_n.$
  • Figure 4: Based on $1000$ samples from $g$, centered and rescaled histograms based on $300$ repetitions, $V^{(\underline{x},\overline{x})}_n$ and their difference compared with respective limiting distributions $L_x$, $N(0,\sigma^2_{\underline{x},\overline{x}})$ and $W$.
  • Figure :

Theorems & Definitions (25)

  • Proposition 1: Construction estimators
  • Remark 1
  • Theorem 1: Asymptotics
  • Proposition 2: LAN expansion
  • Theorem 2: LAM & Convolution
  • Lemma 1
  • proof
  • proof : Proof of Proposition \ref{['prop: construction estimators']}
  • Lemma 2
  • proof
  • ...and 15 more