Maximum Likelihood and Quasi-Likelihood for Nonlinear Exponential Family Regression Models

Abstract

Linear and nonlinear exponential family and quasi-likelihood regression models form a class of models with a structure that invites using one algorithmic framework to compute parameter estimates and regression diagnostics. This framework extends our work on nonlinear least squares; it includes iteratively reweighted least squares but also encompasses secant updates for part of the Hessian matrix of the likelihood or quasi-likelihood function along with tests for when to use this information. The framework also provides basic machinery for computing “leave one out”-style regression diagnostics. We describe the framework, discuss some implementation details, and present some numerical experience.

Key Words:

• Generalized linear model
• Logistic regression
• Nonlinear regression
• Poisson regression
• Quasi—Newton method
• Regression diagnostic
• Robust regression