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

TL;DR

This paper analyzes the asymptotic properties of maximum likelihood estimators in generalized linear models when data is censored with 1-bit measurements, providing theoretical insights and practical applications.

Contribution

It establishes regularity conditions for MLE consistency and normality under 1-bit censoring in GLMs, deriving Fisher information matrices for censored data.

Findings

Fisher information matrices for censored and uncensored data derived.

Conditions for MLE consistency and asymptotic normality established.

Application to Gaussian and Poisson models with 1-bit measurements.

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.