Given data from a sample of noisy curves, we consider a nonlinear parametric regression model with unknown model function. An iterative algorithm for estimating individual parameters as well as the model function is introduced under the assumption of a certain shape invariance: the individual regression curves are obtained from a common shape function by linear transformations of the axes. Our algorithm is based on least-squares methods for parameter estimation and on nonparametric kernel methods for curve estimation. Asymptotic distributions are derived for the individual parameter estimators as well as for the estimator of the shape function. An application to human growth data illustrates the method.
"Model Estimation in Nonlinear Regression Under Shape Invariance." Ann. Statist. 23 (2) 551 - 570, April, 1995. https://doi.org/10.1214/aos/1176324535