Discussiones Mathematicae Probability and Statistics 20(1) (2000) 97-114

Roland Gautier and Luc Pronzato

*CNRS/Université de Nice Sophia Antipolis, France*

The optimal experiment for estimating the parameters of a nonlinear regression model
usually depends on the value of these parameters, hence the problem of designing
experiments that are *robust* with respect to parameter uncertainty. Sequential
designpermits to adapt the experiment to the value of the parameters, and can thus be
considered as a robust design procedure. By designing theexperiments sequentially, one
introduces a feedback of information, and thus dynamics, into the design procedure.
Several sequential schemes, corresponding to different control policies, are considered.
The optimal one corresponds to closed-loop control, and is solution of a stochastic
dynamic-programming problem, which is extremely difficult to solve. A suboptimal strategy
is proposed, which relies ona normal approximation of the future posterior of θ, independent of future observations. The design criterion obtained
involves several mathematical expectations, which are approximated by Laplace method.
Finally, stochastic approximation algorithms are also suggested to determine (sub)optimal
sequential experiments without having to compute expectations.

**Keywords and phrases**: active control, adaptive control, certainty equivalence,
closed-loop control, dynamic programming, open-loop feedback, optimal design, sequential
design.

**1991 Mathematics Subject Classifications**: Primary 62L05, 62K05; Secondary 49L20,
49N35.

[1] | R. Bellman, Dynamic Programming, Princeton University Press, Princeton, N. J.,
1957. |

[2] | K. Chaloner and I. Verdinelli, Bayesian experimental design: a review,
Statistical Science 10 (3) (1995), 273-304. |

[3] | R. Gautier and L. Pronzato, Sequential design and active control,
NewDevelopments and Applications in Experimental Design (N. Flournoy, W. F. Rosenberger
and W. K. Wong, eds.), IMS Lecture Notes 34 (1998), 138-151. |

[4] | C. Kulcsár, L. Pronzato and E. Walter, Optimal experimental design and therapeutic
drug monitoring, Int. Journal of Biomedical Computing 36 (1994), 95-101. |

[5] | C. Kulcsár, L. Pronzato and E. Walter, Dual control of linearly parameterized
models via prediction of posterior densities, European J. of Control 2 (1996),
135-143. |

[6] | H. Kushner and G. Yin, Stochastic Approximation Algorithms and Applications,
Springer, Heidelberg 1997. |

[7] | J. Pilz, Bayesian Estimation and Experimental Design in Linear Regression Models,
vol. 55 Teubner-Texte zur Mathematik, Leipzig, 1983 (also Wiley, New York 1991). |

[8] | L. Pronzato, C. Kulcsár and E. Walter, An actively adaptive control policy for
linear models, IEEE Trans. Autom. Cont. 41 (1996), 855-858. |

[9] | L. Pronzato and E. Walter, Robust experiment design via stochastic approximation,
Mathematical Biosciences 75 (1985), 103-120. |

[10] | L. Pronzato, E. Walter and C. Kulcsár, A dynamical-system approach to sequential
design, pp. 11-24 in: Model-Oriented Data Analysis III, Proceedings MODA3, St
Petersburg, May 1992 (W. G. Müller, H. P. Wynn and A. A. Zhigljavsky, eds.), Physica
Verlag, Heidelberg. |

[11] | W. J. Runggaldier, Concepts of optimality in stochastic control, pp. 101-114
in: Reliability and Decision (R. Barlow, et al., ed.), Elsevier, Amsterdam. |

[12] | M. Tanner, Tools for Statistical Inference Methods for Exploration of Posterior
Distributions and Likelihood Functions, Springer, Heidelberg 1993. |

[13] | L. Tierney and J. Kadane, Accurate approximations for posterior moments and
marginal densities, Journal of the American Statistical Association 81 (393)
(1986), 82-86. |

[14] | S. Zacks, Problems and approaches in design of experiments for estimation and
testing in nonlinear models, pp. 209-223 in: Multivariate Analysis IV, (P.
Krishnaiah, ed.), North Holland, Amsterdam 1977. |

Received 10 December 1998

Revised 24 April 1999