Abstract
Adaptive optimal design with a cost constraint is considered, both for LS estimation in nonlinear regression and ML estimation in Bernoulli-type experiments, with possible applications in clinical trials. We obtain the strong consistency of the estimators for designs over a finite space, both when the cost level is fixed (and the adaptive design converges to an optimum constrained design) and when the objective is to minimize the cost. Moreover, the asymptotic normality of the estimators is obtained in the first situation, with an asymptotic covariance matrix given by the inverse of the usual information matrix, calculated as if the design were not constructed sequentially.
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References
Antognini, A. B. and A. Giovagnoli (2005). On the large sample optimality of sequential designs for comparing two or more treatments. Sequential Analysis 24, 205–217.
Chaudhuri, P. and P. Mykland (1993). Nonlinear experiments: optimal design and inference based likelihood. Journal of the American Statistical Association 88(422), 538–546.
Chaudhuri, P. and P. Mykland (1995). On efficiently designing of nonlinear experiments. Statistica Sinica 5, 421–440.
Cook, R. D. and V. Fedorov (1995). Constrained optimization of experimental design (invited discussion paper). Statistics 26, 129–178.
Cook, R. D. and W. Wong (1994). On the equivalence between constrained and compound optimal designs. Journal of the American Statistical Association 89(426), 687–692.
Dragalin, V. and V. Fedorov (2006). Adaptive designs for dose-finding based on efficacy-toxicity response. Journal of Statistical Planning and Inference 136, 1800–1823.
Dragalin, V., V. Fedorov, and Y. Wu (2008). Adaptive designs for selecting drug combinations based on efficacy-toxicity response. Journal of Statistical Planning and Inference 138, 352–373.
Fedorov, V. and P. Hackl (1997). Model-Oriented Design of Experiments. Berlin: Springer.
Ford, I. and S. Silvey (1980). A sequentially constructed design for estimating a nonlinear parametric function. Biometrika 67(2), 381–388.
Hu, I. (1998). On sequential designs in nonlinear problems. Biometrika 85(2), 496–503.
Lai, T. (1994). Asymptotic properties of nonlinear least squares estimates in stochastic regression models. Annals of Statistics 22(4), 1917–1930.
Lai, T. and H. Robbins (1978). Adaptive design in regression and control. Proc. Nat. Acad. Sci. USA 75(2), 586–587.
Mikulecká, J. (1983). On a hybrid experimental design. Kybernetika 19(1), 1–14.
Müller, W. and B. Pötscher (1992). Batch sequential design for a nonlinear estimation problem. In V. Fedorov, W. Müller, and I. Vuchkov (Eds.), Model-Oriented Data Analysis II, Proceedings 2nd IIASA Workshop, St Kyrik (Bulgaria), May 1990, pp. 77–87. Heidelberg: Physica Verlag.
Pronzato, L. (2000). Adaptive optimisation and D-optimum experimental design. Annals of Statistics 28(6), 1743–1761.
Pronzato, L. (2009a). Asymptotic properties of nonlinear estimates in stochastic models with finite design space. Statistics & Probability Letters 79, 2307–2313. DOI: 10.1016/j.spl.2009.07.025.
Pronzato, L. (2009b). One-step ahead adaptiveD-optimal design on a finite design space is asymptotically optimal. Metrika. (to appear, DOI: 10.1007/s00184-008-0227-y).
Pronzato, L. (2010). Penalized optimal designs for dose-finding. Journal of Statistical Planning and Inference 140, 283–296. DOI: 10.1016/j.jspi.2009.07.012.
Wu, C. (1985). Asymptotic inference from sequential design in a nonlinear situation. Biometrika 72(3), 553–558.
Wynn, H. P. (1970). The sequential generation of D-optimum experimental designs. Annals of Math. Stat. 41, 1655–1664.
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Pronzato, L. (2010). Asymptotic Properties of Adaptive Penalized Optimal Designs over a Finite Space. In: Giovagnoli, A., Atkinson, A., Torsney, B., May, C. (eds) mODa 9 – Advances in Model-Oriented Design and Analysis. Contributions to Statistics. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2410-0_22
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DOI: https://doi.org/10.1007/978-3-7908-2410-0_22
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