New optimal design criteria for regression models with asymmetric errors

• New optimal design criteria are proposed based on the second-order least squares estimator.
• Transformation invariance and symmetry properties of the new criteria are investigated.
• All the results can be applied to both linear and nonlinear regression models.
• New optimal designs are obtained for some polynomial and trigonometric models.

Optimal regression designs are usually constructed by minimizing some scalar functions of the covariance matrix of the ordinary least squares estimator. However, when the error distribution is not symmetric, the second-order least squares estimator is more efficient than the ordinary least squares estimator. Thus we propose new design criteria to construct optimal regression designs based on the second-order least squares estimator. Transformation invariance and symmetry properties of the new criteria are investigated, and sufficient conditions are derived to check for these properties of D-optimal designs. The results can be applied to both linear and nonlinear regression models. Several examples are given for polynomial, trigonometric and exponential regression models, and new designs are obtained.