Compared with the ordinary least squares, the second-order least squares is a more efficient estimation method when the error distribution in a regression model is asymmetric. This paper is concerned with the problem of optimal regression designs based on the second-order least squares estimator under \(I_L\)-optimality which emphasizes the designs to achieve reliable prediction from the fitted regression models. A general equivalence theorem for \(I_L\)-optimality is established and used to check \(I_L\)-optimality of designs. Invariant properties with respect to model reparameterization and linear transformation are also obtained. Several examples are given to illustrate the usefulness of these results.

与普通最小二乘法相比，当回归模型中的误差分布不对称时，二阶最小二乘是一种更有效的估计方法。本文关注的是基于\（I_L \）- optimality下基于二阶最小二乘估计的最优回归设计问题，该最优回归设计强调了从拟合回归模型中获得可靠预测的设计。建立了\（I_L \）-最优性的一般等价定理，并将其用于检查设计的\（I_L \）-最优性。还获得了关于模型重新参数化和线性变换的不变性质。给出了几个例子来说明这些结果的有用性。