On the existence of the least squares estimate in nonlinear growth curve models of exponential type

A general criterion for the existence of a global minimizer of a continuous function on a noncompact set is developed. Criteria for the existence of the least squares estimate in some popular nonlinear growth curve models of exponential type are derived: the quasilinear regression model, two- and three-parameter exponential model, modified exponential model, Gompertz curve, and logistic model. The concept of the ''existence level'', as the minimum of the sum of squares on the boundary of the parameter set, is introduced. Simple procedures for checking whether a nonlinear least squares estimate exists, and suitable initial starting values for particular growth curve models are presented. These concepts and derived criteria are illustrated using the logistic model on a real life biomedical example of mouse tumor growth.