The Connection Between Methods of Estimation in Implicit and Explicit Nonlinear Models

Abstract

This paper establishes the connection between the methods of Britt and Luecke (1973) and Dolby (1972) for obtaining maximum likelihood (ML) estimates of the parameters of nonlinear functional relationships. Britt and Luecke postulated an implicit functional relationship and used Lagrange multipliers and Taylor series linearization of the fitted function to maximize the likelihood subject to equality constraints. Dolby postulated an explicit functional relationship and used the method of scoring to solve iteratively the unconstrained ML equations. It is shown here that if the implicit relationship can be written in explicit form then the two methods are equivalent. This result extends that of Ratkowsky and Dolby (1975) who showed the equivalence of Taylor series linearization and scoring for parameters in the context of simple nonlinear regression.