Bayesian and maximin A-optimal designs for spline regression models with unknown knots

Optimal designs for spline regression models with multiple unknown knots are investigated using A-optimality, Bayesian and maximin criteria. Locally A-optimal designs are constructed, which depend on the true values of the knots. In practice, the knots are never known exactly, but it may be reasonable to assume a prior distribution for the knots. Using a finite discrete distribution for the knots, we propose to apply a Bayesian or maximin efficiency criterion to construct optimal designs. Several theoretical results are derived, including the number of support points in A-optimal designs, and a necessary and sufficient condition for Bayesian A-optimal designs. As it is challenging to find those optimal designs analytically, we propose to use a numerical method for computing them. The key to using the numerical method is to transform the design problems into convex optimization problems. The method is effective and fast to compute optimal designs on discrete design spaces. A-optimal, Bayesian A-optimal and maximin efficiency optimal designs are presented for quadratic and cubic spline models on discrete design spaces, and their features are discussed and compared.