Study of the excess property of the L-optimal design for the Laible model

For a rational two-dimensional nonlinear in parameters Laible model used in analytical chemistry, the problem of constructing L-optimal designs is investigated. It is shown that there are two types of optimal designs for this model: saturated (i. e., designs with the number of support points equal to the number of model parameters) and excess (i. e., designs with the number of support points greater than the number of model parameters) and that with some homothetic transformations of the design space, locally L-optimal designs can change the type from saturated to excess and vice versa. An analytical solution to the problem of finding the dependence between the number of the optimal design support points and the values of the model parameters based on the application of a functional approach is proposed. The L-efficiency of D-optimal designs is investigated.