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SOME PROPERTIES AND IMPROVEMENTS OF

THE SADDLEPOINT APPROXIMATION IN

NONLINEAR REGRESSION

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ANDREJ PÁZMAN

*Department of Probability and Statistics, Faculty of Mathematics and Physics,*

Comenius University, 842 15 Bratislava, Slovak Republic
(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 *chi*^{2} 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.

**Source**
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