D-optimal designs based on the second-order least squares estimator

Abstract

When the error distribution in a regression model is asymmetric, the second-order least squares estimator (SLSE) is more efficient than the ordinary least squares estimator. This result motivated the research in Gao and Zhou (J Stat Plan Inference 149:140–151, 2014), where A-optimal and D-optimal design criteria based on the SLSE were proposed and various design properties were studied. In this paper, we continue to investigate the optimal designs based on the SLSE and derive new results for the D-optimal designs. Using convex optimization techniques and moment theories, we can construct D-optimal designs for univariate polynomial and trigonometric regression models on any closed interval. Several theoretical results are obtained. The methodology is quite general. It can be applied to reduced polynomial models, reduced trigonometric models, and other regression models. It can also be extended to A-optimal designs based on the SLSE.

Appendix: Proofs

Proof of Theorem 1

From (4), we have

$$\begin{aligned} \text{ det }(V(\hat{\varvec{\theta }}_{ SLS}))= & {} \frac{\sigma ^{2q} (1-t)^q }{1-t ~\varvec{g}_1^\top \varvec{G}_2^{-1} \varvec{g}_1} \text{ det }\left( \varvec{G}_2^{-1} \right) \\= & {} \frac{\sigma ^{2q} (1-t)^q }{\left( 1-t ~\varvec{g}_1^\top \varvec{G}_2^{-1} \varvec{g}_1\right) \text{ det }\left( \varvec{G}_2 \right) } \\= & {} \frac{\sigma ^{2q} (1-t)^q }{\text{ det }\left( \mathbf A \right) }. \end{aligned}$$

$\square$

Proof of Theorem 3

• (i) For the $\xi _D^{ SLS}$, it is clear that $\phi (\mu _1^*, \ldots , \mu _{2q}^*) < \infty$, which implies that $\det \left( \mathbf{A}(t, \mu _1^*, \ldots , \mu _{2q}^*) \right) > 0$. By (5), we have $\det \left( \varvec{G}_2\right) > 0$, so the rank of $\varvec{G}_2$ is $q$. For model (7), since $\varvec{G}_2 = \sum _{i=1}^N p_i \mathbf{f}(x_i^*) \mathbf{f}^\top (x_i^*)$, we must have $N \ge q$, which means that there must be at least $q$ support points. From Curto and Fialkow (1991) and Step (2 - i) in Algorithm I, there are at most $q+1$ support points. Thus the number of support points is either $q$ or $q+1$.

• (ii) For a symmetric design space $S=[-c, c]$, Gao and Zhou (2014) have proved that there exists a symmetric D-optimal design $\xi _D^{ SLS}$. We prove by contradiction that the support points include the two boundary points. Suppose the ordered support points of a symmetric $\xi _D^{ SLS}$ are $x_1^* < x_2^* < \cdots < x_N^*$, and $|x_1^*|=x_N^* < c$. Define $\nu =\frac{c}{x_N^*}$, then we have $\nu > 1$ and all the points $\nu x_1^*, \nu x_2^*,\ldots , \nu x_N^*$ are still in the design space. Define a distribution $\xi _D^+$ on $S=[-c, c]$ as

  $$\begin{aligned} \xi _D^{+} = \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \nu x_1^* &{} \nu x_2^* &{} \cdots &{} \nu x_N^* \\ p_1 &{} p_2 &{} \cdots &{} p_N \end{array} \right) , \end{aligned}$$

  and its moments are $\mu _j^+= \nu ^j \mu _j^*$, $j=1, \ldots , 2q$. By (8), it is easy to verify that

  $$\begin{aligned} \det \left( \mathbf{A}(t, \mu _1^+, \ldots , \mu _{2q}^+) \right)= & {} \nu ^{q(q+1)} \det \left( \mathbf{A}(t, \mu _1^*, \ldots , \mu _{2q}^*) \right) \\> & {} \det \left( \mathbf{A}(t, \mu _1^*, \ldots , \mu _{2q}^*) \right) , \end{aligned}$$

  which implies that $\phi (\mu _1^+, \ldots , \mu _{2q}^+) < \phi (\mu _1^*, \ldots , \mu _{2q}^*)$. This is a contradiction to the fact that the $\xi _D^{ SLS}$ minimizes $\phi (\mu _1, \ldots , \mu _{2q})$. Therefore, the support points of $\xi _D^{ SLS}$ must include the two boundary points $-c$ and $c$. The results in (iii) and (iv) can be proved similarly to the result in (ii), and the proof is omitted.

$\square$