研究1比特测量下广义线性模型的极大似然估计渐近性质。
Generalized Linear Models with 1-Bit Measurements: Asymptotics of the Maximum Likelihood Estimator
- 基于指数族分布和线性预测，分析1比特数据下的最大似然估计。
- 证明了参数估计在大样本下一致且渐近正态，给出信息矩阵表达式。
- 适用于均值方差未知的高斯模型或泊松均值估计等实际场景。
本文建立了从截断数据中获得的多参数最大似然估计（MLE）一致性与渐近正态性的正则条件，其中截断机制为1比特测量。假设未截断数据服从指数族分布，自然参数为预测变量的线性组合，即广义线性模型（GLM）。分析中推导了截断与未截断数据的Fisher信息矩阵，用于量化截断影响并评估MLE性能。该框架可覆盖多种实际应用，如均值与方差均未知的高斯模型，以及均值未知的泊松模型，所获结果可用于具体场景的统计推断。
原文摘要 · Abstract (English)
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.
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