Different approaches used in designs an experiment for nonlinear regression models are locally optimal design, minimax, maximinbayesian and bayesian. All these approaches correspond to different levels of a priori information about the parameters of the model.
The bayesian design of construction experimental designs for one-parametric exponential regression model are considered in this article. Five bayesianDubov’s D-functionals are used as a optimally criterion. Necessary and sufficient optimal conditions generalizing classic theorem of Kifer-Volfovich to bayesian case are obtained in Grigoriev, Dubov, Fedorov, Atkinson and Donev works for these functionals.
In the one-dimensional case, these five functionals reduce to three functionals that generate three classes of bayesian designs. In the general case, the construction of such designs requires the assignment of an a priory distribution satisfying the corresponding regularity conditions. In this paper, a uniform distribution is considered as such a distribution.
For the Dubovfunctionals under consideration and for a given a priori distribution, one-point bayesian designs are constructed and their branch points are found, that is, parameters of the support of the a priori distribution for which the spectrum of the one-point design is replenished by the second point. With a further increase in the diameter of the carrier, the spectrum of the optimal design is expanded due to the branching of the second point of the spectrum, and so on. The problem of finding the second branch point in the paper is not considered.
To verify bayesian designs for optimality, the Chaloner-Larnz optimality theorem is used, from which it follows that in the general case the bayesian spectrum may contain a number of points exceeding the number of unknown parameters of the model.
In this article for example of one-parameter exponential model shows that three concerned Dubov’s functional (in general case-all five functional) possess different level of self-descriptiveness. For more information criterion being among three mentioned Dubov’s functional, first bifurcation point is taken to infinity. It is mean that bayesian optimal design is one-pointed for any diameter bearer of own a prior distribution.
A.S. Grunyashin
1. Grigor'ev Yu.D., Denisov V.I., Korablev V.I. Baiesovskie plany eksperimenta pri nelineinoi parametrizatsii [Bayesian experiment plans for nonlinear parameterization]. Primenenie EVM v optimal'nom planirovanii i proektirovanii [Computer application in optimal planning and design]. Novosibirsk, Novosibirsk Electrotechnical Institute Publ., 1974, pp. 3–9.
2. Grigor'ev Yu.D. Nekotoryevoprosypostroeniyaplanoveksperimentaprinelineinoiparametrizatsii.Diss. kand.tekhn. nauk[Some questions of constructing experimental plans for nonlinear parameterization.PhD eng. sci. diss.].Novosibirsk, 1975.157 p.
3. Fedorov V.V. Teoriyaoptimal'nogoeksperimenta [Theory of optimal experiments]. Moscow, Nauka Publ., 1971. 312 p.
4. Grigor'ev Yu.D. Metodyoptimal'nogoplanirovaniyaeksperimenta: lineinyemodeli[Methods of optimal experiment planning: linear models]. St. Petersburg, Lan' Publ., 2015. 320 p.
5. Dubov E.L. D-optimal'nyeplanypribaiesovskompodkhodedlyanelineinoiparametrizatsii[D-optimal plans for the Bayesian approach for nonlinear parameterization].Regressionnyeeksperimenty: (planirovanieianaliz)[Regression experiments (planning and analysis)]. Ed. by V.V. Nalimov.Moscow, Moskovskiiuniversitet Publ., 1977, pp. 103–111.
6. Dubov E.L, Baiesovskieoptimal'nyeplany v nelineinykhzadachakh[Bayesian optimal plans for nonlinear problems]. Voprosykibernetiki.Lineinayainelineinayaparametrizatsiya v zadachakhplanirovaniyaeksperimenta[Problems of cybernetics.Linear and nonlinear parametrization in problems of experimental design].Ed. by V.V. Nalimov, V.V. Fedorov. Moscow, 1981, pp. 27–30.
7. Fedorov V.V. Aktivnyeregressionnyeeksperimenty[Active regression experiments]. Matematicheskiemetodyplanirovaniyaeksperimenta[Mathematical methods of experiment planning].Ed. by V.V. Penenko. Novosibirsk, Nauka Publ., 1981, pp. 19–73.
8. Chaloner K., Larntz K. Optimal Bayesian experimental design applied to logistic regression experiments. Journal of Statistical Planning and Inference, 1989, vol. 21 (2), pp. 191–208.
9. Chaloner K., Larntz K. Bayesian design for accelerated life testing. Journal of Statistical Planning and Inference, 1992, vol. 33 (2), pp. 245–260.
10. Chaloner K. A note on optimal Bayesian design for nonlinear problems.Journal of Statistical Planning and Inference, 1993, vol. 37 (2), pp. 229–236.
11. Dette H., Sperlich S.A note on Bayesian D-optimal designs for a generalization of the exponential growth model. South African Statistical Journal, 1994, vol. 28, pp. 103–117.
12. Dette H., Neugebauer H.M. Bayesian optimal one point designs for one parameter nonlinear models. Journal of Statistical Planning and Inference, 1996, vol. 52 (1), pp. 17–31.
13. Dette H., Neugebauer H.M. Bayesian D-optimal designs for exponential regression models. Journal of Statistical Planning and Inference, 1997, vol. 60 (2), pp. 331–349.
14. Brass D., Dette H. On the number of support points of maximin and Bayesian D-optimal designs in nonlinear regression models. The Annals of Statistics, 2007, vol. 35 (2), pp. 772–792.
15. Staroselskiy Yu.M. Issledovaniebaiesovskikh D-optimal'nykhplanovdlyadrobno-ratsional'nykhmodelei [Study of Bayesian D-optimal designs for rational vodels].Vestnik Sankt-Peterburgskogouniversiteta.Seriya 10.Prikladnayamatematika.Informatika.Protsessyupravleniya – Vestnik of Saint Petersburg university. Applied mathematics.Computer science. Control processes, 2008, iss. 3, pp. 98–105.
16. Starosel'skii Yu.M. Issledovanieoptimal'nykhplanoveksperimentadlyanelineinykhpoparametramregressionnykhmodelei.Dis. kand.phis.-mat. Nauk [Investigation of optimal experimental designs for regression models that are non-linear in parameters.PhD phys. and math.sci. diss.].St. Petersburg, 2008.109 p.
17. Atkinson A.C., Donev A.N. Optimal experimental designs. Oxford, Clarendon Press, 1992.352 p.
