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September, 1977 Maximum Likelihood Estimates in Exponential Response Models
Shelby J. Haberman
Ann. Statist. 5(5): 815-841 (September, 1977). DOI: 10.1214/aos/1176343941


Exponential response models are a generalization of logit models for quantal responses and of regression models for normal data. In an exponential response model, $\{F(\theta): \theta \in \Theta\}$ is an exponential family of distributions with natural parameter $\theta$ and natural parameter space $\Theta \subset V$, where $V$ is a finite-dimensional vector space. A finite number of independent observations $S_i, i \in I$, are given, where for $i \in I, S_i$ has distribution $F(\theta_i)$. It is assumed that $\mathbf{\theta} = \{\theta_i: \mathbf{i} \in \mathbf{I}\}$ is contained in a linear subspace. Properties of maximum likelihood estimates $\hat\mathbf{\theta}$ of $\mathbf{\theta}$ are explored. Maximum likelihood equations and necessary and sufficient conditions for existence of $\hat\mathbf{\theta}$ are provided. Asymptotic properties of $\hat\mathbf{\theta}$ are considered for cases in which the number of elements in $I$ becomes large. Results are illustrated by use of the Rasch model for educational testing.


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Shelby J. Haberman. "Maximum Likelihood Estimates in Exponential Response Models." Ann. Statist. 5 (5) 815 - 841, September, 1977.


Published: September, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0368.62019
MathSciNet: MR501540
Digital Object Identifier: 10.1214/aos/1176343941

Primary: 62F10
Secondary: 62E20

Keywords: Asymptotic theory , exponential family , logit model , maximum likelihood estimation , quantal response , Rasch model

Rights: Copyright © 1977 Institute of Mathematical Statistics


Vol.5 • No. 5 • September, 1977
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