Dispersion Models

Introduction

A dispersion model, denoted Y ∼ DM(μ, σ ²), is a two-parameter family of distributions with probability density functions on ℝ of the form

$$f(y;\mu,{\sigma }^{2}) = a(y;{\sigma }^{2})\exp \left \{- \frac{1} {2{\sigma }^{2}}d(y;\mu )\right \}.$$

(1)

Here μ and σ ² are real parameters with domain (μ, σ ²) ∈ Ω ×ℝ ₊ (Ω being an interval), called the position and dispersion parameters, respectively. Also a and d are suitable functions such that (1) is a probability density function for all parameter values. In particular, d is assumed to be a unit deviance, satisfying d(μ; μ) = 0 for μ ∈ Ω and d(y; μ) > 0 for y≠μ. Dispersion models were introduced by Sweeting (1981) and Jørgensen (1983; 1987b) who extended the analysis of deviance for generalized linear models in the sense of Nelder and Wedderburn (1972) to non-linear regression models with error distribution DM(μ, σ ²).

In many cases, the unit deviance d is regular, meaning that it is twice continuously differentiable and ∂...