Numerical Methods for Finding A-optimal Designs Analytically

Abstract

The traditional way in statistics to find optimal designs for regression models is an analytical approach. Technical conditions that may be restrictive in practice are sometimes imposed to obtain the analytical results. Even then, the mathematical technique is invariably not amendable to find an optimal design under a different criterion or for the same criterion with a slightly changed model, suggesting that developing flexible and effective algorithms to search for the optimum is very useful. In particular, numerical results from an algorithm can be helpful to find analytical descriptions of optimal designs. As an example, particle swarm optimization has been shown to be quite effective for finding optimal designs for hard design problems and this paper demonstrates how its output can be used to find new analytic $A$-optimal approximate designs for the Gamma and inverse Gaussian models, each with the inverse link function. The methodology is quite general and may be applied to find analytical $A$-optimal designs for other models, like the Poisson model with the log link function, or other types of optimal designs.