Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonparametric Estimation in Uniform Deconvolution and Interval Censoring

Piet Groeneboom, Geurt Jongbloed

TL;DR

This paper studies nonparametric estimation of the distribution function $F_0$ in the uniform deconvolution model $S=U+V$ with $V$ uniform on $[0,1]$, focusing on the dichotomy determined by whether $F_0(1)=1$. It connects the problem to interval censoring, showing that $F_0(1)=1$ yields a current-status equivalent, while $F_0(1)<1$ corresponds to IC-$m$ with $m=⌈M⌉$, leading to iterative MLE procedures. The authors derive a cube-root-type local limit in the unit-support case and formulate two competing conjectures for the general case, then introduce a mixed uniform deconvolution model with random exposure length $E$, deriving a Brownian-motion–type local limit with a constant $c_E$ and comparing the two conjectures for the fixed-case variance. They also show that certain smooth functionals of $F_0$, such as the mean or density, admit $\sqrt{n}$-consistent inference via smooth-functional theory, with explicit score representations; and they provide simulations validating the conjectures and illustrating differences between the fixed and mixed models. Overall, the work clarifies the asymptotic regimes of uniform deconvolution, links deconvolution to interval censoring, and outlines concrete conjectures and methodology for future rigorous validation.

Abstract

In the uniform deconvolution problem one is interested in estimating the distribution function $F_0$ of a nonnegative random variable, based on a sample with additive uniform noise. A peculiar and not well understood phenomenon of the nonparametric maximum likelihood estimator in this setting is the dichotomy between the situations where $F_0(1)=1$ and $F_0(1)<1$. If $F_0(1)=1$, the MLE can be computed in a straightforward way and its asymptotic pointwise behavior can be derived using the connection to the so-called current status problem. However, if $F_0(1)<1$, one needs an iterative procedure to compute it and the asymptotic pointwise behavior of the nonparametric maximum likelihood estimator is not known. In this paper we describe the problem, connect it to interval censoring problems and a more general model studied in Groeneboom (2024) to state two competing naturally occurring conjectures for the case $F_0(1)<1$. Asymptotic arguments related to smooth functional theory and extensive simulations lead us to to bet on one of these two conjectures.

Nonparametric Estimation in Uniform Deconvolution and Interval Censoring

TL;DR

This paper studies nonparametric estimation of the distribution function in the uniform deconvolution model with uniform on , focusing on the dichotomy determined by whether . It connects the problem to interval censoring, showing that yields a current-status equivalent, while corresponds to IC- with , leading to iterative MLE procedures. The authors derive a cube-root-type local limit in the unit-support case and formulate two competing conjectures for the general case, then introduce a mixed uniform deconvolution model with random exposure length , deriving a Brownian-motion–type local limit with a constant and comparing the two conjectures for the fixed-case variance. They also show that certain smooth functionals of , such as the mean or density, admit -consistent inference via smooth-functional theory, with explicit score representations; and they provide simulations validating the conjectures and illustrating differences between the fixed and mixed models. Overall, the work clarifies the asymptotic regimes of uniform deconvolution, links deconvolution to interval censoring, and outlines concrete conjectures and methodology for future rigorous validation.

Abstract

In the uniform deconvolution problem one is interested in estimating the distribution function of a nonnegative random variable, based on a sample with additive uniform noise. A peculiar and not well understood phenomenon of the nonparametric maximum likelihood estimator in this setting is the dichotomy between the situations where and . If , the MLE can be computed in a straightforward way and its asymptotic pointwise behavior can be derived using the connection to the so-called current status problem. However, if , one needs an iterative procedure to compute it and the asymptotic pointwise behavior of the nonparametric maximum likelihood estimator is not known. In this paper we describe the problem, connect it to interval censoring problems and a more general model studied in Groeneboom (2024) to state two competing naturally occurring conjectures for the case . Asymptotic arguments related to smooth functional theory and extensive simulations lead us to to bet on one of these two conjectures.

Paper Structure

This paper contains 10 sections, 7 theorems, 68 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Uniform deconvolution and interval censoring
  3. Case $F_0(1)=1$
  4. Case $F_0(1)<1$
  5. The mixed model
  6. Interlude; examples of applications of smooth functional theory in the fixed model
  7. Simulations and asymptotic considerations
  8. Concluding remarks
  9. Appendix
  10. Score equations

Key Result

Theorem 2.1

Let $F_0$ be differentiable on $(a,b),\,0<a<b<1$ with a continuous positive derivative $f_0(t)$ for $t\in(a,b)$, where $[a,b]$ is the support of $f_0$. Let $\hat{F}_n$ be the nonparametric MLE of $F_0$. Then, for $t_0\in(a,b)$: where $W$ is two-sided Brownian motion on $\mathbb R$, originating from zero.

Figures (4)

  • Figure 1: The score function $\theta_{F_0}$ for the truncated exponential distribution, given by (\ref{['score_function_truncated_exp']}).
  • Figure 2: Simulated variances, times $n^{2/3}$, for the mixed model of $\hat{F}_n(t_i)$, for $t_i=0.1,0.2,\dots,1.9$ (blue solid curve, linearly interpolated between values at the $t_i$), compared with the asymptotic values (\ref{['sigma_t']}) of Theorem 4.1 in piet_EJS:24 (red, dashed) for $E_i$ uniform on $[0.5,1.5]$. The simulated variances are based on $10,000$ simulations of samples of size $n=10,000$ for (a) $F_0$ the truncated exponential distribution function on $[0,2]$ and (b) $F_0$ the uniform distribution function on $[0,2]$. The blue dashed curves are the corresponding asymptotic variance curves of Conjecture \ref{['conjecture:limit_df']} for the fixed model.
  • Figure 3: Simulated variances, times $n^{2/3}$, for the fixed model of $\hat{F}_n(t_i)$, for $t_i=0.1,0.2,\dots,1.9$ (blue solid curve, linearly interpolated between values at the $t_i$), compared with the asymptotic values of Conjecture \ref{['conjecture:limit_df']} (blue, dashed). The simulated variances are based on $10,000$ simulations of samples of size $n=10,000$ for (a) $F_0$ the truncated exponential distribution function on $[0,2]$ and (b) $F_0$ the uniform distribution function on $[0,2]$. The red dashed curves are the corresponding asymptotic variance curves of Theorem 4.1 in piet_EJS:24 for the mixed model and $E_i$ uniform on $[0.5,1.5]$.
  • Figure 4: Simulated variances, times $n^{2/3}$, for the fixed model of $\hat{F}_n(t_i)$, for $t_i=0.1,0.2,\dots,1.9$ (black solid curve, linearly interpolated between values at the $t_i$), compared with the asymptotic values of Conjecture \ref{['conjecture:limit_df']} (blue, dashed) and the conjecture that would follow from Theorem 4.1 in piet_EJS:24, ignoring the conditions (purple, dotted), for (a) $F_0$ the truncated exponential distribution function on $[0,2]$ and (b) $F_0$ the uniform distribution function on $[0,2]$. The simulated variances are based on $10,000$ simulations of samples of size $n=10,000$.

Theorems & Definitions (14)

  • Theorem 2.1
  • Definition 2.1
  • Lemma 2.1
  • Theorem 3.1
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • Theorem 4.1
  • Lemma 5.1
  • proof
  • ...and 4 more