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Generalized Linear Models with 1-Bit Measurements: Asymptotics of the Maximum Likelihood Estimator

Jaimin Shah, Martina Cardone, Cynthia Rush, Alex Dytso

TL;DR

The paper addresses parameter estimation from 1-bit censored measurements within a generalized linear model for exponential-family data. It derives a Fisher information framework for both censored and uncensored data and proves mild regularity conditions under which the maximum likelihood estimator is consistent and asymptotically normal, with asymptotic covariance given by the inverse of the limiting FIM $oldsymbol{J}$. The results are applied to Gaussian (unknown mean and variance) and Poisson (unknown mean) models, illustrating how censoring impacts information and how thresholds can be chosen to preserve efficiency. This work provides a unifying theory for 1-bit censored GLMs and offers practical insights for sensor fusion, radar, and photon-counting scenarios where 1-bit measurements are prevalent.

Abstract

This work establishes regularity conditions for consistency and asymptotic normality of the multiple parameter maximum likelihood estimator(MLE) from censored data, where the censoring mechanism is in the form of $1$-bit measurements. The underlying distribution of the uncensored data is assumed to belong to the exponential family, with natural parameters expressed as a linear combination of the predictors, known as generalized linear model (GLM). As part of the analysis, the Fisher information matrix is also derived for both censored and uncensored data, which helps to quantify the impact of censoring and assess the performance of the MLE. The choice of GLM allows one to consider a variety of practical examples where 1-bit estimation is of interest. In particular, it is shown how the derived results can be used to analyze two practically relevant scenarios: the Gaussian model with both unknown mean and variance, and the Poisson model with an unknown mean.

Generalized Linear Models with 1-Bit Measurements: Asymptotics of the Maximum Likelihood Estimator

TL;DR

The paper addresses parameter estimation from 1-bit censored measurements within a generalized linear model for exponential-family data. It derives a Fisher information framework for both censored and uncensored data and proves mild regularity conditions under which the maximum likelihood estimator is consistent and asymptotically normal, with asymptotic covariance given by the inverse of the limiting FIM . The results are applied to Gaussian (unknown mean and variance) and Poisson (unknown mean) models, illustrating how censoring impacts information and how thresholds can be chosen to preserve efficiency. This work provides a unifying theory for 1-bit censored GLMs and offers practical insights for sensor fusion, radar, and photon-counting scenarios where 1-bit measurements are prevalent.

Abstract

This work establishes regularity conditions for consistency and asymptotic normality of the multiple parameter maximum likelihood estimator(MLE) from censored data, where the censoring mechanism is in the form of -bit measurements. The underlying distribution of the uncensored data is assumed to belong to the exponential family, with natural parameters expressed as a linear combination of the predictors, known as generalized linear model (GLM). As part of the analysis, the Fisher information matrix is also derived for both censored and uncensored data, which helps to quantify the impact of censoring and assess the performance of the MLE. The choice of GLM allows one to consider a variety of practical examples where 1-bit estimation is of interest. In particular, it is shown how the derived results can be used to analyze two practically relevant scenarios: the Gaussian model with both unknown mean and variance, and the Poisson model with an unknown mean.
Paper Structure (20 sections, 4 theorems, 35 equations, 1 figure)

This paper contains 20 sections, 4 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Relevant literature
  3. Our Contributions and Paper Outline
  4. Problem Formulation
  5. Main Result
  6. Fisher Information
  7. Consistency and Asymptotic Normality of the MLE
  8. Examples
  9. Example 1: Gaussian Distribution
  10. Example 2: Poisson Distribution
  11. CONCLUSION
  12. Proof of (5)
  13. Proof of Proposition 1
  14. Proof of Theorem 1
  15. Derivation of the FIM for the Gaussian Distribution
  16. ...and 5 more sections

Key Result

Proposition 1

The FIM of estimating the true parameter vector $\boldsymbol{\theta}_0$ from the censored data $\{B_i\}_{i=1}^n$ is given by

Figures (1)

  • Figure 1: Case 3: Unknown mean and variance; $X_i \sim \mathcal{N}(2,1)$ for all $i \in [1:n]$.

Theorems & Definitions (7)

  • Proposition 1
  • proof
  • Theorem 1
  • Remark 1
  • Proposition 2
  • Lemma 1
  • proof