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(θ): θ ∈ Θ} is an exponential family of distributions with natural parameter θ and natural parameter space $\Theta \subset V$, where V is a finite-dimensional vector space. A finite number of independent observations Si, i ∈ I, are given, where for i ∈ I, Si has distribution F(θi). It is assumed that θ = {θi: i ∈ I} is contained in a linear subspace. Properties of maximum likelihood estimates θ̂ of θ are explored. Maximum likelihood equations and necessary and sufficient conditions for existence of θ̂ are provided. Asymptotic properties of θ̂ 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.]