Designs in nonlinear regression by stochastic minimization of functionals of the mean square error matrix

We reconsider and extend the method of designing nonlinear experiments presented in Pázman and Pronzato (J. Statist. Plann. Inference 33 (1992) 385). The approach is based on the probability density of the LS estimators, and takes into account the boundary of the parameter space. The idea is to express the elements of the mean square error matrix (MSE) of the LS estimators as integrals of the density, express optimality criteria as functions of MSE, and minimize them by stochastic optimization. In the present paper we include prior knowledge about some parameters, we derive improved approximations of the density of estimators, and we consider not only linear optimality criteria like generalized A-optimality, but also D  - and LsLs-optimality criteria with an integer s. Of basic importance is the use of an accelerated method of stochastic optimization (MSO). This together with the important progress in computing allowed us to elaborate a realistic algorithm. Its performance is demonstrated by the search of a generalized D-optimum design in the 4-parameter growth curve model of microbiological experiments presented by Baranyi and Roberts (Internat. J. Food Microbiol. 23 (1994) 277; 26 (1995) 199).