The Annals of Statistics

Maximum Likelihood Estimates in Exponential Response Models

Shelby J. Haberman

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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.

Article information

Ann. Statist., Volume 5, Number 5 (1977), 815-841.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F10: Point estimation
Secondary: 62E20: Asymptotic distribution theory

Exponential family maximum likelihood estimation quantal response logit model Rasch model asymptotic theory


Haberman, Shelby J. Maximum Likelihood Estimates in Exponential Response Models. Ann. Statist. 5 (1977), no. 5, 815--841. doi:10.1214/aos/1176343941.

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