Invited discussion paper small-sample distributional properties of nonlinear regression estimators (a geometric approach)

The paper is mainly a survey of the topic how to approximate the probability density of the parameter estimator in a nonlinear regression model. A short presentation of the geometry of the model and a heuristic discussion of the model and a heuristic discussion of the “irregularities” of the estimates are given. In the model with Gaussian errors we present the asymptotic normal approximation, the approximationby the second order Edgeworth expansion, a conditional density of BARNDORFF-NIELSEN, and mainly the approximation called “flat” or “saddlepoint” approximation, which will be shown to have several interesting properties. Further, we present the possibility of improving the approximation in some models, the extension of the approximation to some cases of nongaussian errors, and besides the maximum likelihood estimator we consider also the weighted least-squares estimator, with the weights not depending on the error concariance matrix.

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