Densities of Nonlinear Functions of the Nonlinear Least-Squares Estimator

Abstract We present a new method for the construction of the density of the least-squares estimator of a nonlinear function of the parameters in nonlinear regression. Important applications might correspond to marginal densities of the parameters and densities of predicted responses. One-dimensional densities can easily be plotted, which provides an efficient tool for judging the quality of the identification with respect to the intended application. The problem is of special importance when the number of observations is small, so that the classical asymptotic normal approximation for the density of the estimator (which corresponds to a linearization) cannot be used. Two approximations of different levels of accuracy are obtained. A numerical example is presented. A comparison is made with a more general method of the literature, which is shown to be ineffective in the context of nonlinear regression.