$I_L$-optimal designs for regression models under the second-order least squares estimator

Abstract

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.

Appendix: The derivation of the analytical results in Example 2

Let $\eta _j=\int _{{\mathcal {X}}}x^j\xi (dx)$ be the jth moment of x based on the design measure $\xi$. We have $g_1(\xi )=\begin{pmatrix}\eta _1\\ \eta _2\end{pmatrix}$ and $G_2(\xi )=\begin{pmatrix}\eta _2&{}\eta _3\\ \eta _3&{}\eta _4\end{pmatrix}$. From Theorem 4 the criterion function $\psi _L(\xi )$ in (6) is invariant with respect to the reflection of both x and z, then we just need to focus on a symmetric design measure, say ${\bar{\xi }}$, on ${\mathcal {X}}=[-1,1]$ in order to construct $I_L$-optimal designs. For all the moments $\eta _j$, we have $\eta _1=\eta _3=0$ under symmetric designs, and the even moments of any design measure satisfy $0\le \eta _2^2\le \eta _4\le \eta _2\le 1$ from Dette and Studden (1997). The following is an explicit derivation for the case $L=0$, the others can be derived similarly and are omitted for saving space.

For $L=0$, the criterion function $\psi _L(\cdot )$ in (6) under symmetric designs simplifies to

$$\begin{aligned} \psi _L({\bar{\xi }})=\exp \left\{ \int _0^1\log \left( \frac{z^2}{\eta _2}+\frac{z^4}{\eta _4-t\eta _2^2} \right) dz\right\} , \end{aligned}$$

then the $I_0$-optimal design $\xi _0^*$ can be achieved through solving the following optimization problem:

$$\begin{aligned} \min \limits _{{\bar{\xi }}} \quad&\log \left( \frac{1}{\eta _2}+\frac{1}{\eta _4-t\eta _2^2}\right) +2\sqrt{\frac{\eta _4-t\eta _2^2}{\eta _2}}\arctan \left( \sqrt{\frac{\eta _2}{\eta _4-t\eta _2^2}}\right) \\ \text{ s.t. }\quad&0\le \eta _2^2\le \eta _4\le \eta _2\le 1. \end{aligned}$$

Denote the above-mentioned objective function by ${\widetilde{\psi }}_0(\eta _2,\eta _4)$, a straightforward calculation yields $\partial {\widetilde{\psi }}_0(\eta _2,\eta _4)/\partial \eta _4\le 0$, which implies ${\widetilde{\psi }}_0(\eta _2,\eta _4)$ decreases in $\eta _4$ for a fixed $\eta _2$ and attains its minimum at $\eta _4=\eta _2$. It follows that the design $\xi _0^*$ belongs to the set of symmetric designs of the following form

$$\begin{aligned} {\bar{\xi }}= \begin{Bmatrix} -1&0&1 \\ \frac{\omega }{2}&1-\omega&\frac{\omega }{2} \\ \end{Bmatrix} \end{aligned}$$

for some $\omega \in (0, 1]$, and the objective function ${\widetilde{\psi }}_0(\eta _2,\eta _4)$ can be rewritten as

$$\begin{aligned} {\widetilde{\psi }}_0(\omega )=\log (2-\omega t)-\log (\omega -\omega ^2 t)+2(1-\omega t)^{\frac{1}{2}}\arctan \left[ (1-\omega t)^{-\frac{1}{2}}\right] . \end{aligned}$$

Therefore, ${\widetilde{\psi }}_0(\omega )$ is minimized at $\omega =\min \{1,\omega _0^*\}$ where $\omega _0^*$ is the root of $\partial {\widetilde{\psi }}_0(\omega )/\partial \omega =0$, which is equal to (15). This completes the derivation.