Maximum Likelihood Estimates in Exponential Response Models

Abstract

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.