Convergence and consistency results for self-modeling nonlinear regression. (English) Zbl 0725.62060

Kneip, Alois; Gasser, Theo

Ann. Stat. 16, No. 1, 82-112 (1988).

Summary: This paper is concerned with parametric regression models of the form \[ Y_{ij}=f(t_{ij},\theta_ i)+error,\quad i=1,...,n,\quad j=1,...,T_ i, \] where the continuous function f may depend nonlinearly on the known regressors \(t_{ij}\) and the unknown parameter vectors \(\theta_ i\). 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 \(\theta_ i\) by making use of the prior information. An iterative algorithm simplifying computation of the estimates is presented, and for \(\min \{n,T_ 1,...,T_ n\}\to \infty\) conditions for strong uniform consistency of the resulting estimators of f and strong consistency of the estimators of \(\theta_ i\) are established. Some examples illustrating the method are included.

MSC:

62J02	General nonlinear regression
62F10	Point estimation
62G07	Density estimation

Keywords:

model selection; method of sieves; preliminary exploratory data analysis; identifiability; simultaneous estimation; prior information; iterative algorithm; strong uniform consistency; strong consistency