Maximum Likelihood Estimates in Exponential Response Models

Exponential response models are a generalization of logit models for quantal responses and of regression models for normal data. In an exponential response model, (θ): θ ∈ Θ\ is an exponential family of distributions with natural parameter θ and natural parameter space Θ ⊂ 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 θ of θ are explored. Maximum likelihood equations and necessary and sufficient conditions for existence of θ are provided. Asymptotic properties of θ 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.