Convergence and Consistency Results for Self-Modeling Nonlinear Regression

This paper is concerned with parametric regression models of the form Yᵢⱼ = f(tᵢⱼ, θᵢ) + error, i = 1, …, n, j = 1, …, Tᵢ, where the continuous function f may depend nonlinearly on the known regressors tᵢⱼ and the unknown parameter vectors θᵢ. The assumption of an a priori known f is dropped and replaced by the requirement that qualitative information about the structure of the model is available or can be generated by a preliminary exploratory data analysis. This framework--allowing both f and the individual parameter vectors to be unknown--necessitates a detailed discussion of identifiability of model and parameters. A method is then proposed for the simultaneous estimation of f and θᵢ by making use of the prior information. An iterative algorithm simplifying computation of the estimates is presented, and for min, T₁, …, Tₙ\ → ∞ conditions for strong uniform consistency of the resulting estimators of f and strong consistency of the estimators of θᵢ are established. Some examples illustrating the method are included.