Maximum likelihood computation based on the Fisher scoring and Gauss–Newton quadratic approximations

Abstract

The Fisher scoring and Gauss–Newton methods are two known methods for maximum likelihood computation. This paper provides a generalization for each method in a unified manner so that they can be used for some difficult maximum likelihood computation, when, for example, there exist constraints on the parameters. A generalized method does not use directly the Newton-type iteration formulas of these methods, but, instead, uses the corresponding quadratic functions transformed from them. It proceeds by repeatedly approximating the log-likelihood function with the quadratic functions in the neighborhoods of the current iterates and optimizing each quadratic function within the parameter space. It is shown that each quadratic function has a weighted linear regression formulation, which can be conveniently solved. This generalization also extends the applicability of the Fisher scoring method to situations when the expected Fisher information matrices are unavailable in closed form. Fast computation can generally be anticipated, owing to their small rates of convergence and a rapid solution of each linear regression problem. While the generalized Gauss–Newton method may sometimes suffer for the so-called large residual problem, the generalized Fisher scoring method has performed consistently well in the numerical experiments we conducted.