(Received June 9, 1997; revised December 9, 1997)
Abstract. We summarize properties of the saddlepoint approximation of the density of the maximum likelihood estimator in nonlinear regression with normal errors: accuracy, range of validity, equivariance. We give a geometric insight into the accuracy of the saddlepoint density for finite samples. The role of the Riemannian curvature tensor in the whole investigation of the properties is demonstrated. By adding terms containing this tensor we improve the saddlepoint approximation. When this tensor is zero, or when the number of observations is large, we have pivotal, independent, and chi2 distributed variables, like in a linear model. Consequences for experimental design or for constructions of confidence regions are discussed.
Key words and phrases: Least squares, curvatures, differential geometry, distribution of estimators, confidence regions.