Skip to main content
SpringerLink
Log in

One-step ahead adaptive D-optimal design on a finite design space is asymptotically optimal

  • Published:
Metrika Aims and scope Submit manuscript

Abstract

We study the consistency of parameter estimators in adaptive designs generated by a one-step ahead D-optimal algorithm. We show that when the design space is finite, under mild conditions the least-squares estimator in a nonlinear regression model is strongly consistent and the information matrix evaluated at the current estimated value of the parameters strongly converges to the D-optimal matrix for the unknown true value of the parameters. A similar property is shown to hold for maximum-likelihood estimation in Bernoulli trials (dose–response experiments). Some examples are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Atkinson A, Donev A (1992) Optimum experimental design. Oxford University Press, NY, USA

    Google Scholar 

  • Chaloner K, Larntz K (1989) Optimal Bayesian design applied to logistic regression experiments. J Stat Plan Inference 21: 191–208

    Article  MATH  MathSciNet  Google Scholar 

  • Chaloner K, Verdinelli I (1995) Bayesian experimental design: a review. Stat Sci 10(3): 273–304

    Article  MATH  MathSciNet  Google Scholar 

  • Chaudhuri P, Mykland P (1993) Nonlinear experiments: optimal design and inference based likelihood. J Am Stat Assoc 88(422): 538–546

    Article  MATH  MathSciNet  Google Scholar 

  • Chaudhuri P, Mykland P (1995) On efficiently designing of nonlinear experiments. Statistica Sinica 5: 421–440

    MATH  MathSciNet  Google Scholar 

  • Downing D, Fedorov V, Leonov S (2001) Extracting information from the variance function: optimal design. In: Atkinson A, Hackl P, Müller W(eds) mODa6—advances in model-oriented design and analysis. Physica Verlag, Heidelberg, pp 45–52

    Google Scholar 

  • Dragalin V, Fedorov V (2006) Adaptive designs for dose-finding based on efficacy–toxicity response. J Stat Plan Inference 136: 1800–1823

    Article  MATH  MathSciNet  Google Scholar 

  • Fedorov V (1972) Theory of optimal experiments. Academic Press, New York

    Google Scholar 

  • Fedorov V (1980) Convex design theory. Math Operationsforsch. Statist Ser Statistics 11(3): 403–413

    MATH  Google Scholar 

  • Ford I, Silvey S (1980) A sequentially constructed design for estimating a nonlinear parametric function. Biometrika 67(2): 381–388

    Article  MATH  MathSciNet  Google Scholar 

  • Ford I, Titterington D, Wu C (1985) Inference and sequential design. Biometrika 72(3): 545–551

    Article  MathSciNet  Google Scholar 

  • Hu I (1998) On sequential designs in nonlinear problems. Biometrika 85(2): 496–503

    Article  MATH  MathSciNet  Google Scholar 

  • Jennrich R (1969) Asymptotic properties of nonlinear least squares estimation. Ann Math Stat 40: 633–643

    Article  MATH  MathSciNet  Google Scholar 

  • Kiefer J, Wolfowitz J (1960) The equivalence of two extremum problems. Can J Math 12: 363–366

    MATH  MathSciNet  Google Scholar 

  • Lai T (1994) Asymptotic properties of nonlinear least squares estimates in stochastic regression models. Ann Stat 22(4): 1917–1930

    Article  MATH  Google Scholar 

  • Lai T, Wei C (1982) Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. Ann Stat 10(1): 154–166

    Article  MathSciNet  Google Scholar 

  • Melas V (1978) Optimal designs for exponential regressions. Math Operationsforsch und Statist, Ser Statistics 9: 753–768

    MathSciNet  Google Scholar 

  • Müller W, Pötscher B (1992) Batch sequential design for a nonlinear estimation problem. In: Fedorov V, Müller W, Vuchkov I (eds) Model-Oriented Data Analysis II. Proceedings 2nd IIASA Workshop, St Kyrik (Bulgaria), May 1990. Physica Verlag, Heidelberg, pp 77–87

  • Pázman A, Pronzato L (2004) Simultaneous choice of design and estimator in nonlinear regression with parameterized variance. In: Di Bucchianico, A, Läuter H, Wynn H (eds) mODa’7—Advances in model-oriented design and analysis. Proceedings of the 7th international workshop, Heeze (Netherlands). Physica Verlag, Heidelberg, pp 117–124

  • Pázman A, Pronzato L (2007) Quantile and probability-level criteria for nonlinear experimental design. In: López-Fidalgo J, Rodríguez-Díaz J, Torsney B (eds) mODa’8—Advances in model-oriented design and analysis. Proceedings of the 8th international workshop, Almagro (Spain). Physica Verlag, Heidelberg, pp 157–164

  • Pronzato L, Walter E (1985) Robust experiment design via stochastic approximation. Math Biosci 75: 103–120

    Article  MATH  MathSciNet  Google Scholar 

  • Pronzato L, Walter E (1988) Robust experiment design via maximin optimization. Math Biosci 89: 161–176

    Article  MATH  MathSciNet  Google Scholar 

  • Rosenberger W, Flournoy N, Durham S (1997) Asymptotic normality of maximum likelihood estimators for multiparameter response-driven designs. J Stat Plan Inference 60: 69–76

    Article  MATH  MathSciNet  Google Scholar 

  • Shiryaev A (1996) Probability. Springer, Berlin

    Google Scholar 

  • Wu C (1981) Asymptotic theory of nonlinear least squares estimation. Ann Stat 9(3): 501–513

    Article  MATH  Google Scholar 

  • Wu C (1985) Asymptotic inference from sequential design in a nonlinear situation. Biometrika 72(3): 553–558

    Article  MATH  MathSciNet  Google Scholar 

  • Wynn H (1970) The sequential generation of D-optimum experimental designs. Ann Math Stat 41: 1655–1664

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire I3S, CNRS, Université de Nice-Sophia Antipolis, Bât Euclide, Les Algorithmes, 2000 route des lucioles, BP 121, 06903, Sophia Antipolis Cedex, France

    Luc Pronzato

Authors
  1. Luc Pronzato
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luc Pronzato.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pronzato, L. One-step ahead adaptive D-optimal design on a finite design space is asymptotically optimal. Metrika 71, 219–238 (2010). https://doi.org/10.1007/s00184-008-0227-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00184-008-0227-y

Keywords

Search

Navigation