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 χ2 distributed variables, like in a linear model. Consequences for experimental design or for constructions of confidence regions are discussed.
Similar content being viewed by others
REFERENCES
Amari, S. I. (1985). Differential-geometrical Methods in Statistics, Lecture Notes in Statist., No. 28, Springer, Berlin.
Barndorff-Nielsen, O. E. (1980). Conditionality resolutions, Biometrika, 67, 293–310.
Barndorff-Nielsen, O. E. and Cox, D. R. (1979). Edgeworth and saddle-point approximations with statistical applications, J. Roy Statist. Soc. Ser B, 41, 279–312.
Bates, D. M. and Watts, D. G. (1980). Relative curvature measures of nonlinearity, J. Roy. Statist. Soc. Ser. B, 42, 1–25.
Bates, D. M. and Watts, D. G. (1988). Nonlinear Regression Analysis and its Applications, Wiley, New York.
Clarke, G. P. Y. (1980). Moments of the least-squares estimators in nonlinear regression models, J. Roy. Statist. Soc. Ser. B, 42, 227–237.
Eisenhart, L. P. (1960). Riemannian Geometry, Princeton University Press, Princeton.
Hougaard, P. (1985). Saddlepoint approximations for curved exponential families, Statist. Probab. Lett., 3, 161–166.
Hougaard, P. (1986). Covariance stabilizing transformations in nonlinear regression, Scand. J. Statist., 13, 207–210.
Hougaard, P. (1995). Nonlinear regression and curved exponential families. Improvement of the approximation to asymptotic distribution, Metrika, 42, 191–202.
Jensen, J. L. (1995). Saddlepoint Approximations, Oxford University Press, New York.
Pázman, A. (1984). Probability distribution of the multivariate least squares estimates, Kybernetika, 20, 209–230.
Pázman, A. (1990). Almost exact distributions of estimators I and II, Statistics, 21, 9–19 and 21–32.
Pázman, A. (1991). Pivotal variables and confidence regions in flat nonlinear regression models with unknown ±, Statistics, 22, 177–189.
Pázman, A. (1993a). Higher dimensional nonlinear regression—A statistical use of the Riemannian curvature tensor, Statistics, 25, 17–25.
Pázman, A. (1993b). Nonlinear Statistical Models, Kluwer, Dordrecht.
Pázman, A. and Pronzato, L. (1992). Nonlinear experimental design based on the distribution of estimators, J. Statist. Plann. Inference, 33, 385–402.
Seber, G. A. F. and Wild, C. J. (1989). Nonlinear Regression, Wiley, New York.
Skovgaard, I. M. (1985). Large deviation approximations for maximum likelihood estimators, Probab. Math. Statist., 6, 89–107.
Skovgaard, I. M. (1990). On the density of minimum contrast estimators, Ann. Statist., 18, 779–789.
Author information
Authors and Affiliations
About this article
Cite this article
Pázman, A. Some Properties and Improvements of the Saddlepoint Approximation in Nonlinear Regression. Annals of the Institute of Statistical Mathematics 51, 463–478 (1999). https://doi.org/10.1023/A:1003946021224
Issue Date:
DOI: https://doi.org/10.1023/A:1003946021224