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Estimation and inference in error-in-operator model

Vladimir Spokoiny

TL;DR

The paper develops a finite-sample semiparametric framework for Estimation and Inference in the Error-in-Operator model by enlarging the parameter space to include the operator $A$ and its image $\boldsymbol{z}$, and by regularizing through ridge or truncation penalties. It establishes strong local concavity of the penalized extended likelihood, derives Fisher and Wilks-type expansions for the profiled estimator, and provides explicit risk bounds in terms of effective dimension and regularization, clarifying when ridge regularization or spectral truncation yields optimal rates. The key contributions include a detailed analysis of the phase transition between ridge and truncation depending on operator smoothness, a minimax-like rate discussion in mildly ill-posed inverse problems, and guidance on how to control estimation risk via penalty design in high-dimensional EiO settings with random design or instrumental variables. The results have practical implications for high-dimensional inverse problems where the forward operator is uncertain, enabling near-oracle performance and informing model-reduction strategies in applied statistical inference.

Abstract

Many statistical problems can be reduced to a linear inverse problem in which only a noisy version of the operator is available. Particular examples include random design regression, deconvolution problem, instrumental variable regression, functional data analysis, error-in-variable regression, drift estimation in stochastic diffusion, and many others. The pragmatic plug-in approach can be well justified in the classical asymptotic setup with a growing sample size. However, recent developments in high dimensional inference reveal some new features of this problem. In high dimensional linear regression with a random design, the plug-in approach is questionable but the use of a simple ridge penalization yields a benign overfitting phenomenon; see \cite{baLoLu2020}, \cite{ChMo2022}, \cite{NoPuSp2024}. This paper revisits the general Error-in-Operator problem for finite samples and high dimension of the source and image spaces. A particular focus is on the choice of a proper regularization. We show that a simple ridge penalty (Tikhonov regularization) works properly in the case when the operator is more regular than the signal. In the opposite case, some model reduction technique like spectral truncation should be applied.

Estimation and inference in error-in-operator model

TL;DR

The paper develops a finite-sample semiparametric framework for Estimation and Inference in the Error-in-Operator model by enlarging the parameter space to include the operator and its image , and by regularizing through ridge or truncation penalties. It establishes strong local concavity of the penalized extended likelihood, derives Fisher and Wilks-type expansions for the profiled estimator, and provides explicit risk bounds in terms of effective dimension and regularization, clarifying when ridge regularization or spectral truncation yields optimal rates. The key contributions include a detailed analysis of the phase transition between ridge and truncation depending on operator smoothness, a minimax-like rate discussion in mildly ill-posed inverse problems, and guidance on how to control estimation risk via penalty design in high-dimensional EiO settings with random design or instrumental variables. The results have practical implications for high-dimensional inverse problems where the forward operator is uncertain, enabling near-oracle performance and informing model-reduction strategies in applied statistical inference.

Abstract

Many statistical problems can be reduced to a linear inverse problem in which only a noisy version of the operator is available. Particular examples include random design regression, deconvolution problem, instrumental variable regression, functional data analysis, error-in-variable regression, drift estimation in stochastic diffusion, and many others. The pragmatic plug-in approach can be well justified in the classical asymptotic setup with a growing sample size. However, recent developments in high dimensional inference reveal some new features of this problem. In high dimensional linear regression with a random design, the plug-in approach is questionable but the use of a simple ridge penalization yields a benign overfitting phenomenon; see \cite{baLoLu2020}, \cite{ChMo2022}, \cite{NoPuSp2024}. This paper revisits the general Error-in-Operator problem for finite samples and high dimension of the source and image spaces. A particular focus is on the choice of a proper regularization. We show that a simple ridge penalty (Tikhonov regularization) works properly in the case when the operator is more regular than the signal. In the opposite case, some model reduction technique like spectral truncation should be applied.

Paper Structure

This paper contains 32 sections, 21 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Example: Random design regression
  3. Example: Instrumental variable regression
  4. This paper contribution
  5. Outlook
  6. Organization of the paper
  7. Theoretical study for the error-in-operator model
  8. Main definitions and model conditions
  9. Full dimensional information matrix
  10. Local concavity and local smoothness
  11. Conditions on the errors
  12. Loss and risk of estimation
  13. Stochastic term. Effective and critical dimension
  14. A bound on the penalization bias and the estimation risk
  15. Risk bounds and choice of penalty
  16. ...and 17 more sections

Key Result

Theorem 2.1

Assume EU2ref for $\nabla \zeta = (0,\boldsymbol{\omega},\mu \, \boldsymbol{U})$. Let $D^{2}$ satisfy tfeuhfuwr365vdyge34def and let, with $N = \lambda_{\min}(D^{2})$ and $R = \delta_{0} \mu \sqrt{N}$ for $\delta_{0} \leq 1/10$, the set $\varUpsilon^{\circ}$ from rughur47478ytfg84g6gh contain $\bol with $\varkappa = 2$. Then the following statements hold for any linear mapping $Q$ on $\mathbbmsl{

Theorems & Definitions (36)

  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • Proposition 3.4
  • ...and 26 more