Properties of optimal regression designs under the second-order least squares estimator

We investigate properties of optimal designs under the second-order least squares estimator (SLSE) for linear and nonlinear regression models. First we derive equivalence theorems for optimal designs under the SLSE. We then obtain the number of support points in A-, c- and D-optimal designs analytically for several models. Using a generalized scale invariance concept we also study the scale invariance property of D-optimal designs. In addition, numerical algorithms are discussed for finding optimal designs. The results are quite general and can be applied for various linear and nonlinear models. Several applications are presented, including results for fractional polynomial, spline regression and trigonometric regression models.