Criteria Based on the Small-Sample Precision of the LS Estimator

  • Luc Pronzato
  • Andrej Pázman
Part of the Lecture Notes in Statistics book series (LNS, volume 212)


This chapter deals with designs with a fixed finite size N (exact designs), of the form X = (x 1, , x N ), possibly with replications; that is, we may have x i = x j for some ij.


Confidence Region Marginal Density Nonlinear Regression Model Riemannian Curvature Tensor Intrinsic Curvature 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Bates, D. and D. Watts (1980). Relative curvature measures of nonlinearity. J. Roy. Statist. Soc. B42, 1–25.Google Scholar
  2. Box, M. (1971). Bias in nonlinear estimation. J. Roy. Statist. Soc. B33, 171–201.Google Scholar
  3. Clarke, G. (1980). Moments of the least-squares estimators in a non-linear regression model. J. Roy. Statist. Soc. B42, 227–237.Google Scholar
  4. Clyde, M. and K. Chaloner (2002). Constrained design strategies for improving normal approximations in nonlinear regression problems. J. Stat. Plann. Inference 104, 175–196.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Efron, B. and D. Hinkley (1978). Assessing the accuracy of the maximum likelihood estimator: observed versus expected Fisher information. Biometrika 65(3), 457–487.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Eisenhart, L. (1960). Riemannian Geometry. Princeton, N.J.: Princeton Univ. Press.Google Scholar
  7. Ermoliev, Y. and R.-B. Wets (Eds.) (1988). Numerical Techniques for Stochastic Optimization Problems. Berlin: Springer.Google Scholar
  8. Firth, D. (1993). Bias reduction of maximum likelihood estimates. Biometrika 80(1), 27–38.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Gauchi, J.-P. and A. Pázman (2006). Designs in nonlinear regression by stochastic minimization of functionals of the mean square error matrix. J. Stat. Plann. Inference 136, 1135–1152.zbMATHCrossRefGoogle Scholar
  10. Gilmour, S. and L. Trinca (2012). Optimum design of experiments for statistical inference (with discussion). J. Roy. Statist. Soc. C61(3), 1–25.Google Scholar
  11. Halperin, M. (1963). Confidence interval estimation in non-linear regression. J. Roy. Statist. Soc. B25, 330–333.Google Scholar
  12. Hamilton, D. and D. Watts (1985). A quadratic design criterion for precise estimation in nonlinear regression models. Technometrics 27, 241–250.MathSciNetzbMATHCrossRefGoogle Scholar
  13. Hamilton, D., D. Watts, and D. Bates (1982). Accounting for intrinsic nonlinearities in nonlinear regression parameter inference regions. Ann. Statist. 10, 386–393.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Hartley, H. (1964). Exact confidence regions for parameters in non-linear regression laws. Biometrika 51(3&4), 347–353.MathSciNetzbMATHGoogle Scholar
  15. Hougaard, P. (1985). Saddlepoint approximations for curved exponential families. Statist. Probab. Lett. 3, 161–166.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Kushner, H. and D. Clark (1978). Stochastic Approximation Methods for Constrained and Unconstrained Systems. Heidelberg: Springer.CrossRefGoogle Scholar
  17. Kushner, H. and G. Yin (1997). Stochastic Approximation Algorithms and Applications. Heidelberg: Springer.zbMATHGoogle Scholar
  18. Lindsay, B. and B. Li (1997). On the second-order optimality of the observed Fisher information. Ann. Statist. 25(5), 2172–2199.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Pázman, A. (1984b). Probability distribution of the multivariate nonlinear least-squares estimates. Kybernetika 20, 209–230.MathSciNetzbMATHGoogle Scholar
  20. Pázman, A. (1990). Small-sample distributional properties of nonlinear regression estimators. a geometric approach (with discussion). Statistics 21(3), 323–367.Google Scholar
  21. Pázman, A. (1992a). A classification of nonlinear regression models and parameter confidence regions. Kybernetika 28(6), 444–453.MathSciNetzbMATHGoogle Scholar
  22. Pázman, A. (1992b). Geometry of the nonlinear regression with prior. Acta Math. Univ. Comenianae LXI, 263–276.Google Scholar
  23. Pázman, A. (1993a). Higher dimensional nonlinear regression — a statistical use of the Riemannian curvature tensor. Statistics 25, 17–28.MathSciNetzbMATHCrossRefGoogle Scholar
  24. Pázman, A. (1993b). Nonlinear Statistical Models. Dordrecht: Kluwer.zbMATHGoogle Scholar
  25. Pázman, A. and L. Pronzato (1992). Nonlinear experimental design based on the distribution of estimators. J. Stat. Plann. Inference 33, 385–402.zbMATHCrossRefGoogle Scholar
  26. Pázman, A. and L. Pronzato (1996). A Dirac function method for densities of nonlinear statistics and for marginal densities in nonlinear regression. Statist. Probab. Lett. 26, 159–167.MathSciNetzbMATHCrossRefGoogle Scholar
  27. Pázman, A. and L. Pronzato (1998). Approximate densities of two bias–corrected nonlinear LS estimators. In A. Atkinson, L. Pronzato, and H. Wynn (Eds.), MODA’5 – Advances in Model–Oriented Data Analysis and Experimental Design, Proc. 5th Int. Workshop, Marseille, 22–26 juin 1998, pp. 145–152. Heidelberg: Physica Verlag.Google Scholar
  28. Pronzato, L. and A. Pázman (1994a). Bias correction in nonlinear regression via two-stages least-squares estimation. In M. Blanke and T. Söderström (Eds.), Prep. 10th IFAC/IFORS Symp. on Identification and System Parameter Estimation, Volume 1, Copenhagen, pp. 137–142. Danish Automation Society.Google Scholar
  29. Pronzato, L. and A. Pázman (1994b). Second-order approximation of the entropy in nonlinear least-squares estimation. Kybernetika 30(2), 187–198. [Erratum 32(1):104, 1996].Google Scholar
  30. Pronzato, L. and A. Pázman (2001). Using densities of estimators to compare pharmacokinetic experiments. Comput. Biol. Med. 31(3), 179–195.CrossRefGoogle Scholar
  31. Pronzato, L. and E. Walter (1985). Robust experiment design via stochastic approximation. Math. Biosci. 75, 103–120.MathSciNetzbMATHCrossRefGoogle Scholar
  32. Ross, G. (1990). Nonlinear estimation. New York: Springer.zbMATHCrossRefGoogle Scholar
  33. Sundaraij, N. (1978). A method for confidence regions for nonlinear models. Austral. J. Statist. 20(3), 270–274.MathSciNetCrossRefGoogle Scholar
  34. Vila, J.-P. (1990). Exact experimental designs via stochastic optimization for nonlinear regression models. In Proc. Compstat, Int. Assoc. for Statistical Computing, pp. 291–296. Heidelberg: Physica Verlag.Google Scholar
  35. Vila, J.-P. and J.-P. Gauchi (2007). Optimal designs based on exact confidence regions for parameter estimation of a nonlinear regression model. J. Stat. Plann. Inference 137, 2935–2953.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Luc Pronzato
    • 1
  • Andrej Pázman
    • 2
  1. 1.French National Center for Scientific Research (CNRS)University of NiceSophia AntipolisFrance
  2. 2.Department of Applied Mathematics and StatisticsComenius UniversityBratislavaSlovakia

Personalised recommendations