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I-robust and D-robust designs on a finite design space

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Abstract

We present and discuss the theory of minimax I- and D-robust designs on a finite design space, and detail three methods for their construction that are new in this context: (i) a numerical search for the optimal parameters in a provably minimax robust parametric class of designs, (ii) a first-order iterative algorithm similar to that of Wynn (Ann Math Stat 5:1655–1664, 1970), and (iii) response-adaptive designs. These designs minimize a loss function, based on the mean squared error of the predicted responses or the parameter estimates, when the regression response is possibly misspecified. The loss function being minimized has first been maximized over a neighbourhood of the approximate and possibly inadequate response being fitted by the experimenter. The methods presented are all vastly more economical, in terms of the computing time required, than previously available algorithms.

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Acknowledgements

This work was carried out with the support of the Natural Sciences and Engineering Research Council of Canada. It has benefited greatly from the insightful comments of two anonymous referees.

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Correspondence to Douglas P. Wiens.

Appendix: Derivations

Appendix: Derivations

Proof of Theorem 1

First note that by (3), \(\varvec{\psi }\) is orthogonal to the columns of \( {\mathbf {Q}}\) and hence lies in the column space of the \(N\times N-p\) matrix \( {\mathbf {Q}}_{\perp }\), whose orthonormal columns span the orthogonal complement \(col\left( {\mathbf {Q}}\right) ^{\perp }\) and hence are orthogonal to those of \({\mathbf {Q}}\). Thus, using (4), \(\varvec{ \psi }=\left( \tau /\sqrt{n}\right) {\mathbf {Q}}_{\perp }{\mathbf {c}}\) for some \( {\mathbf {c}}\) with \(\left\| {\mathbf {c}}\right\| \le 1\). We repeatedly use the identity \({\mathbf {Q}}_{\perp }{\mathbf {Q}}_{\perp }^{\prime }={\mathbf {I}} _{N}-{{\mathbf {Q}}}{{\mathbf {Q}}}^{\prime }\), from which it follows that

$$\begin{aligned} {\mathbf {Q}}^{\prime }{\mathbf {D}}\left( {\varvec{\xi }}\right) {\mathbf {Q}}_{\perp } {\mathbf {Q}}_{\perp }^{\prime }{\mathbf {D}}\left( {\varvec{\xi }}\right) {\mathbf {Q}}={\mathbf {S}}\left( {\varvec{\xi }}\right) -{\mathbf {R}}^{2}{\left( {\varvec{\xi }}\right) .} \end{aligned}$$
(26)

In this notation, and writing the QR-decomposition as \({\mathbf {F}}={{\mathbf {Q}}}{{\mathbf {V}}} \) for a \(p\times p\) triangular \({\mathbf {V}}\), we have \(\mathbf {{\mathbf {M}}= {\mathbf {V}}^{\prime }{\mathbf {R}}\left( {\varvec{\xi }}\right) {\mathbf {V}}}\). Our assumption that \({{\mathbf {M}}}\) be non-singular implies that \({{\mathbf {R}}}\) and \({{\mathbf {V}}}\) are invertible. Also, \(\mathbf { b}=\left( \tau /\sqrt{n}\right) {{\mathbf {V}}^{\prime }Q^{\prime }D\left( {{\varvec{\xi }}}\right) Q}_{\perp }{\mathbf {c}}\) and so

$$\begin{aligned}&{\textsc {mse}}\left[ {\varvec{{\hat{\theta }}}}\right] \\&\quad =\frac{\sigma _{\varepsilon }^{2}+\tau ^{2}}{n}{{\mathbf {V}}}^{-1}\left[ \left( 1-\nu \right) {{\mathbf {R}}}^{-1}{\left( {{\varvec{\xi }}} \right) }\right. \\&\quad \left. +\,\nu {{\mathbf {R}}}^{-1}\left( {\varvec{\xi }}\right) {\mathbf {Q}}^{\prime }{\mathbf {D}}\left( {\varvec{\xi }}\right) {\mathbf {Q}}_{\perp }{{\mathbf {c}}}{{\mathbf {c}}} ^{\prime }{\mathbf {Q}}_{\perp }^{\prime }{\mathbf {D}}\left( {\varvec{\xi }} \right) {\mathbf {Q}}{\mathbf {R}}^{-1}{\left( {\varvec{\xi }}\right) } \right] {{\mathbf {V}}^{\prime }}^{-1}. \end{aligned}$$

For the I-criterion, \(\max _{\psi }{\mathcal {I}}\left( \psi ,{\varvec{\xi }} \right) \) is \(\left( \sigma _{\varepsilon }^{2}+\tau ^{2}\right) /n\) times

$$\begin{aligned}&\max _{\left\| {\mathbf {c}}\right\| \le 1}\left\{ \left( 1-\nu \right) tr{{\mathbf {R}}}^{-1}{\left( {\varvec{\xi }}\right) }\right. \\&\quad \left. +\,\nu {\mathbf {c}}^{\prime }\left[ {\mathbf {Q}}_{\perp }^{\prime }{\mathbf {D}}\left( {\varvec{\xi }}\right) {\mathbf {Q}}{\mathbf {R}}^{-1}{\left( {\varvec{ \xi }}\right) \cdot {{\mathbf {R}}}}^{-1}\left( {\varvec{\xi }}\right) {\mathbf {Q}}^{\prime }{\mathbf {D}}\left( {\varvec{\xi }}\right) {\mathbf {Q}} _{\perp }+{\mathbf {I}}_{N-p}\right] \right\} {\mathbf {c}} \\&\quad =\left( 1-\nu \right) tr{{\mathbf {R}}}^{-1}\mathbf {\left( \varvec{ \xi }\right) }\\&\qquad +\,\nu ch_{\max }\left[ {\mathbf {Q}}_{\perp }^{\prime }{\mathbf {D}} \left( {\varvec{\xi }}\right) {\mathbf {Q}}{\mathbf {R}}^{-1}{\left( {\varvec{\xi }}\right) \cdot {{\mathbf {R}}}^{-1}{\left( {\varvec{\xi }}\right) }Q}^{\prime }{\mathbf {D}}\left( {\varvec{\xi }} \right) {\mathbf {Q}}_{\perp }+{\mathbf {I}}_{N-p}\right] \\&\quad =\left( 1-\nu \right) tr{{\mathbf {R}}}^{-1}{\left( \varvec{ \xi }\right) }\\&\qquad +\, \nu ch_{\max }\left[ {{\mathbf {R}}}^{-1}\left( {\varvec{\xi }}\right) {\mathbf {Q}}^{\prime }{\mathbf {D}}\left( {\varvec{\xi }} \right) {\mathbf {Q}}_{\perp }\cdot {\mathbf {Q}}_{\perp }^{\prime }{\mathbf {D}} \left( {\varvec{\xi }}\right) {\mathbf {Q}}{\mathbf {R}}^{-1}{\left( {\varvec{\xi }}\right) }+{\mathbf {I}}_{p}\right] \\&\quad ={\mathcal {I}}_{\nu }\left( {\varvec{\xi }}\right) , \end{aligned}$$

upon using (26). Here, we make use of the fact that the nonzero eigenvalues of a product \({{\mathbf {A}}}{{\mathbf {A}}}^{\prime }\) coincides with those of \( {\mathbf {A}}^{\prime }{\mathbf {A}}\). For the D-criterion, we have that \( \max _{\psi }{\mathcal {D}}\left( \psi ,{\varvec{\xi }}\right) \) is the p-th root of

$$\begin{aligned}&\left( \frac{\sigma _{\varepsilon }^{2}}{n}\right) ^{p}\left( \frac{\sigma _{\varepsilon }^{2}+\tau ^{2}}{\sigma _{\varepsilon }^{2}\left| {\mathbf {F}}^{\prime }{\mathbf {F}}\right| }\right) \max _{\left\| {\varvec{c}}\right\| \le 1}\nonumber \\&\quad \times \left\{ \frac{1-\nu +\nu {\mathbf {c}}^{\prime }{\mathbf {Q}}_{\perp }^{\prime }{\mathbf {D}}\left( {\varvec{\xi }}\right) {\mathbf {Q}}{\mathbf {R}}^{-1}\left( {\varvec{\xi }}\right) {\mathbf {Q}}^{\prime }{\mathbf {D}}\left( {\varvec{\xi }}\right) {\mathbf {Q}}_{\perp }{\mathbf {c}}}{\left| {{\mathbf {R}}\left( {\varvec{\xi }}\right) }\right| }\right\} \\&=\left( \frac{\sigma _{\varepsilon }^{2}}{n}\right) ^{p}\left( \frac{ \sigma _{\varepsilon }^{2}+\tau ^{2}}{\sigma _{\varepsilon }^{2}\left| {\mathbf {F}}^{\prime }{\mathbf {F}}\right| }\right) \\&\quad \times \left\{ \frac{ 1-\nu +\nu ch_{\max }{\mathbf {Q}}_{\perp }^{\prime }{\mathbf {D}}\left( {\varvec{\xi }}\right) {\mathbf {Q}}{\mathbf {R}}^{-1}\left( \varvec{ \xi }\right) {\mathbf {Q}}^{\prime }{\mathbf {D}}\left( {\varvec{\xi }}\right) {\mathbf {Q}}_{\perp } }{\left| {{\mathbf {R}}\left( {\varvec{\xi }}\right) }\right| } \right\} ; \end{aligned}$$

now using (26) once again completes the proof. \(\square \)

Proof of Theorem 2

A necessary condition for a minimum at \({\varvec{\xi }}_{0}\) is that, for any design \( {\varvec{\xi }}_{1}\) and with \({\varvec{\xi }}_{t}=\left( 1-t\right) {\varvec{\xi }}_{0}+t{\varvec{\xi }}_{1}\),

$$\begin{aligned}&0\le \frac{\mathrm{d}}{\mathrm{d}t}{\mathcal {I}}_{\nu }\left( {\varvec{\xi }}_{t}\right) _{|_{t=0}}=\left( 1-\nu \right) \frac{\mathrm{d}}{\mathrm{d}t}tr{\mathbf {R}}^{-1}\left( {\varvec{\xi }}_{t}\right) _{|_{t=0}}\nonumber \\&\qquad +\,\nu \frac{\mathrm{d}}{\mathrm{d}t}\lambda _{I}\left( {\varvec{\xi }}_{t}\right) _{|_{t=0}}. \end{aligned}$$
(27)

We calculate that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}{\mathbf {R}}^{-1}\left( {\varvec{\xi }}_{t}\right) _{|_{t=0}}= & {} -{\mathbf {R}}^{-1}\left( {\varvec{\xi }}_{0}\right) \left\{ {\mathbf {R}} \left( {\varvec{\xi }}_{1}\right) -{\mathbf {R}}\left( {\varvec{\xi }} _{0}\right) \right\} {\mathbf {R}}^{-1}\left( {\varvec{\xi }}_{0}\right) , \\ \frac{\mathrm{d}}{\mathrm{d}t}{\mathbf {S}}\left( {\varvec{\xi }}_{t}\right) _{|_{t=0}}= & {} 2\left\{ {\mathbf {Q}}^{\prime }{\mathbf {D}}\left( {\varvec{\xi }} _{1}\right) {\mathbf {D}}\left( {\varvec{\xi }}_{0}\right) {\mathbf {Q}}-\mathbf { S}\left( {\varvec{\xi }}_{0}\right) \right\} , \end{aligned}$$

whence

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}tr{\mathbf {R}}^{-1}\left( {\varvec{\xi }}_{t}\right) _{|_{t=0}}= & {} tr\left\{ {\mathbf {R}}^{-1}\left( {\varvec{\xi }}_{0}\right) -{\mathbf {R}} ^{-1}\left( {\varvec{\xi }}_{0}\right) {\mathbf {R}}\left( {\varvec{\xi }} _{1}\right) {\mathbf {R}}^{-1}\left( {\varvec{\xi }}_{0}\right) \right\} , \end{aligned}$$
(28a)
$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}{\mathbf {U}}\left( {\varvec{\xi }}_{t}\right) _{|_{t=0}}= & {} - {\mathbf {R}}^{-1}\left( {\varvec{\xi }}_{0}\right) {\mathbf {R}}\left( {\varvec{\xi }}_{1}\right) {\mathbf {U}}\left( {\varvec{\xi }}_{0}\right) - {\mathbf {U}}\left( {\varvec{\xi }}_{0}\right) {\mathbf {R}}\left( \varvec{ \xi }_{1}\right) {\mathbf {R}}^{-1}\left( {\varvec{\xi }}_{0}\right) \nonumber \\&+\,2{\mathbf {R}}^{-1}\left( {\varvec{\xi }}_{0}\right) {\mathbf {Q}}^{\prime } {\mathbf {D}}\left( {\varvec{\xi }}_{1}\right) {\mathbf {D}}\left( \varvec{ \xi }_{0}\right) {{\mathbf {Q}}}{{\mathbf {R}}}^{-1}\left( {\varvec{\xi }}_{0}\right) . \end{aligned}$$
(28b)

By Theorem 1 of Magnus (1985), \(\lambda _{I}\left( {\varvec{\xi }} _{t}\right) \) is differentiable, with

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\lambda _{I}\left( {\varvec{\xi }}_{t}\right) _{|_{t=0}}= {\mathbf {z}}_{I}^{\prime }({\varvec{\xi }}_{0})\left[ \frac{\mathrm{d}}{\mathrm{d}t}{\mathbf {U}} \left( {\varvec{\xi }}_{t}\right) \right] _{|_{t=0}}{\mathbf {z}}_{I}( {\varvec{\xi }}_{0}). \end{aligned}$$
(29)

Now, let \(\left\{ {\mathbf {q}}_{i}^{\prime }\right\} _{i=1}^{N}\) be the rows of \({\mathbf {Q}}\), and substitute \({\mathbf {R}}\left( {\varvec{\xi }}_{1}\right) =\sum _{i=1}^{N}\xi _{1,i}{\mathbf {q}}_{i}{\mathbf {q}}_{i}^{\prime }\) and \( {\mathbf {Q}}^{\prime }{\mathbf {D}}\left( {\varvec{\xi }}_{1}\right) {\mathbf {D}} \left( {\varvec{\xi }}_{0}\right) {\mathbf {Q}}=\sum _{i=1}^{N}\xi _{1,i}\xi _{0,i}{\mathbf {q}}_{i}{\mathbf {q}}_{i}^{\prime }\) into (27), (28) and (29). One obtains

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}{\mathcal {I}}_{\nu }\left( {\varvec{\xi }}_{t}\right) _{|_{t=0}}=\left( 1-\nu \right) tr{\mathbf {R}}^{-1}\left( {\varvec{\xi }} _{0}\right) -\sum _{i=1}^{N}\xi _{1,i}{\mathbf {T}}_{ii}^{{\mathcal {I}}}\left( {\varvec{\xi }}_{0}\right) . \end{aligned}$$
(30)

Thus (27) is equivalent to

$$\begin{aligned} \sum _{i=1}^{N}\xi _{1,i}\left\{ \frac{{\mathbf {T}}_{ii}^{{\mathcal {I}}}\left( {\varvec{\xi }}_{0}\right) }{\left( 1-\nu \right) tr{\mathbf {R}}^{-1}\left( {\varvec{\xi }}_{0}\right) }-1\right\} \le 0,\text { for all designs } {\varvec{\xi }}_{1}. \end{aligned}$$
(31)

Condition (31) implies that \({\mathbf {T}}_{ii}^{{\mathcal {I}}}\left( {\varvec{\xi }}_{0}\right) / \left[ \left( 1-\nu \right) tr{\mathbf {R}} ^{-1}\left( {\varvec{\xi }}_{0}\right) \right] -1\le 0\) for all i, and then, since

$$\begin{aligned}&\sum _{i=1}^{N}\xi _{0,i}\left\{ \frac{{\mathbf {T}}_{ii}^{{\mathcal {I}}}\left( {\varvec{\xi }}_{0}\right) }{\left( 1-\nu \right) tr{\mathbf {R}}^{-1}\left( {\varvec{\xi }}_{0}\right) }-1\right\} \\&\quad =\frac{tr{\mathbf {D}}\left( {\varvec{\xi }}_{0}\right) {\mathbf {T}}^{{\mathcal {I}}}\left( {\varvec{\xi }} _{0}\right) }{\left( 1-\nu \right) tr{\mathbf {R}}^{-1}\left( {\varvec{\xi }} _{0}\right) }-1=0, \end{aligned}$$

that equality must hold at the points at which \(\xi _{0,i}>0\). \(\square \)

Proof of Theorem 3

The method is the same as for the proof of Theorem 2 and so we mention only the differences. We phrase the necessary condition as

$$\begin{aligned}&0\le \left| {\mathbf {R}}\left( {\varvec{\xi }}_{0}\right) \right| \frac{\mathrm{d}}{\mathrm{d}t}{\mathcal {D}}_{\nu }^{p}\left( {\varvec{\xi }}_{t}\right) _{|_{t=0}} \nonumber \\&\quad =\nu \frac{\mathrm{d}}{\mathrm{d}t}\lambda _{D}\left( {\varvec{\xi }}_{t}\right) _{|_{t=0}}+\left( 1-\nu +\nu \lambda _{D}\left( {\varvec{\xi }}_{0}\right) \right) \nonumber \\&\qquad \times \, \left( p-tr\left( {\mathbf {R}}^{-1}\left( {\varvec{\xi }}_{0}\right) {\mathbf {R}}\left( {\varvec{\xi }}_{1}\right) \right) \right) ; \end{aligned}$$
(32)

here we have used

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\left| {\mathbf {R}}\left( {\varvec{\xi }}_{t}\right) \right| _{|_{t=0}}=\left| {\mathbf {R}}\left( {\varvec{\xi }} _{0}\right) \right| tr\left( {\mathbf {R}}^{-1}\left( {\varvec{\xi }} _{t}\right) \frac{\mathrm{d}}{\mathrm{d}t}{\mathbf {R}}\left( {\varvec{\xi }}_{t}\right) \right) _{|_{t=0}}\\&\quad =\left| {\mathbf {R}}\left( {\varvec{\xi }}_{0}\right) \right| tr\left( {\mathbf {R}}^{-1}\left( {\varvec{\xi }}_{0}\right) \left( {\mathbf {R}}\left( {\varvec{\xi }}_{1}\right) -{\mathbf {R}}\left( {\varvec{\xi }}_{0}\right) \right) \right) . \end{aligned}$$

From the definitions of \({\mathbf {v}}\left( {\varvec{\xi }}\right) \) and \( {\mathbf {w}}\left( {\varvec{\xi }}\right) \), \({\mathbf {v}}^{\prime }\left( {\varvec{\xi }}_{t}\right) {\mathbf {w}}\left( {\varvec{\xi }}_{t}\right) =1 \) and

$$\begin{aligned} \lambda _{D}\left( {\varvec{\xi }}_{t}\right) ={\mathbf {v}}^{\prime }\left( {\varvec{\xi }}_{t}\right) \left[ {\mathbf {U}}\left( {\varvec{\xi }} _{t}\right) -{\mathbf {I}}_{p}\right] {\mathbf {R}}\left( {\varvec{\xi }} _{t}\right) {\mathbf {w}}\left( {\varvec{\xi }}_{t}\right) . \end{aligned}$$

Another appeal to Magnus (1985) gives

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}\lambda _{D}\left( {\varvec{\xi }}_{t}\right) _{|_{t=0}} ={\mathbf {v}}^{\prime }\left( {\varvec{\xi }}_{0}\right) \frac{\mathrm{d}}{\mathrm{d}t}\left[ \left( {\mathbf {U}}\left( {\varvec{\xi }}_{t}\right) -{\mathbf {I}}_{p}\right) {\mathbf {R}}\left( {\varvec{\xi }}_{t}\right) \right] _{|_{t=0}}{\mathbf {w}} \left( {\varvec{\xi }}_{0}\right) \\&={\mathbf {z}}_{D}^{\prime }({\varvec{\xi }}_{0}){\mathbf {R}}^{1/2}\left( {\varvec{\xi }}_{0}\right) \frac{\mathrm{d}}{\mathrm{d}t}\left( \left[ {\mathbf {R}}^{-1}\left( {\varvec{\xi }}_{t}\right) {\mathbf {S}}\left( {\varvec{\xi }}_{t}\right) - {\mathbf {R}}\left( {\varvec{\xi }}_{t}\right) \right] \right) _{|_{t=0}}\\&\quad \times {\mathbf {R}}^{-1/2}\left( {\varvec{\xi }}_{0}\right) {\mathbf {z}}_{D}( {\varvec{\xi }}_{0}). \end{aligned}$$

Then, after a calculation, (32) becomes (9), with equality if \(\xi _{0i}>0\) since

$$\begin{aligned}&\sum _{i=1}^{N}\xi _{0i}\left( \frac{{\mathbf {T}}_{ii}^{{\mathcal {D}}}\left( {\varvec{\xi }}_{0}\right) }{\left( 1-\nu \right) p+\nu \left( p-1\right) \lambda _{D}\left( {\varvec{\xi }}_{0}\right) }-1\right) \\&\quad =\frac{tr\mathbf {D }\left( {\varvec{\xi }}_{0}\right) {\mathbf {T}}^{{\mathcal {D}}}\left( {\varvec{\xi }}_{0}\right) }{\left( 1-\nu \right) p+\nu \left( p-1\right) \lambda _{D}\left( {\varvec{\xi }}_{0}\right) }-1=0. \end{aligned}$$

\(\square \)

Proof of Theorem 4

Note that for a fixed positive definite matrix \({{\mathbf {R}}}\), the class \(\varXi _{{{\mathbf {R}}}}\) of designs \({\varvec{\xi }}\) for which \({{\mathbf {R}}\left( {\varvec{\xi }}\right) ={\mathbf {R}}}\) is convex, and for \( {\varvec{\xi }}\in \varXi _{{{\mathbf {R}}}}\) both \({\mathcal {I}}_{\nu }\left( {\varvec{\xi }}\right) \) and \({\mathcal {D}}_{\nu }\left( \varvec{ \xi }\right) \) are convex. In the case of \({\mathcal {I}}_{\nu }\left( {\varvec{\xi }}\right) \) this follows from the fact that for fixed \({{\mathbf {R}}}\), \({\mathcal {I}}_{\nu }\left( {\varvec{\xi }}\right) \) is a linear function of

$$\begin{aligned} ch_{\max }{\mathbf {U}}\left( {\varvec{\xi }}\right) =ch_{\max }{ {\mathbf {R}}}^{-1}{\mathbf {Q}}^{\prime }{\mathbf {D}}^{2}\left( {\varvec{\xi }} \right) {\mathbf {Q}}{\mathbf {R}}^{-1}; \end{aligned}$$

now note that, with \({\varvec{\xi }}_{t}=\left( 1-t\right) {\varvec{\xi }}_{0}+t{\varvec{\xi }}_{1}\) (\(t\in [0,1]\)), and with \(\preccurlyeq \) denoting the Loewner ordering by positive semi-definiteness,

$$\begin{aligned} {\mathbf {D}}^{2}\left( {\varvec{\xi }}_{t}\right) \preccurlyeq \left( 1-t\right) {\mathbf {D}}^{2}\left( {\varvec{\xi }}_{0}\right) +t{\mathbf {D}} ^{2}\left( {\varvec{\xi }}_{1}\right) . \end{aligned}$$

Thus, \({\mathbf {U}}\left( {\varvec{\xi }}_{t}\right) \preccurlyeq \left( 1-t\right) {\mathbf {U}}\left( {\varvec{\xi }}_{0}\right) +t{\mathbf {U}}\left( {\varvec{\xi }}_{1}\right) \) and (since the maximum eigenvalue of a sum is less than or equal to the sum of the maximum eigenvalues),

$$\begin{aligned} ch_{\max }{\mathbf {U}}\left( {\varvec{\xi }}_{t}\right) \le \left( 1-t\right) ch_{\max }{\mathbf {U}}\left( {\varvec{\xi }}_{0}\right) +tch_{\max }{\mathbf {U}}\left( {\varvec{\xi }}_{1}\right) . \end{aligned}$$

The verification of the convexity of \({\mathcal {D}}_{\nu }\left( \varvec{ \xi }\right) \) for \({\varvec{\xi }}\in \varXi _{{{\mathbf {R}}}}\) is similar.

The losses \({\mathcal {I}}_{\nu }\left( {\varvec{\xi }}\right) \) and \( {\mathcal {D}}_{\nu }\left( {\varvec{\xi }}\right) \) are minimized by minimizing \({\mathcal {I}}_{\nu }\left( {\varvec{\xi }}|{{\mathbf {R}}} \right) =ch_{\max }{\mathbf {U}}\left( {\varvec{\xi }}\right) \) and \(\mathcal { D}_{\nu }\left( {\varvec{\xi }}|{{\mathbf {R}}}\right) =ch_{\max } {{\mathbf {R}}}^{1/2}{\mathbf {U}}\left( {\varvec{\xi }}\right) { {\mathbf {R}}}^{1/2}\), respectively. We first consider the minimization of \( {\mathcal {I}}_{\nu }\left( {\varvec{\xi }}|{{\mathbf {R}}}\right) \). In order that \({\varvec{\xi }}_{0}\) minimize \({\mathcal {I}}_{\nu }\left( {\varvec{\xi }}|{{\mathbf {R}}}\right) \) in \(\varXi _{{{\mathbf {R}}} } \), it is sufficient that the function

$$\begin{aligned} \phi \left( t\right) ={\mathcal {I}}_{\nu }\left( {\varvec{\xi }}_{t}|\mathbf { {\mathbf {R}}}\right) -2tr{\varvec{\varLambda }}{{\mathbf {R}}}\left( {\varvec{\xi }}_{t}\right) -2\alpha {\mathbf {1}}_{N}^{\prime }\varvec{ \xi }_{t} \end{aligned}$$

be minimized unconditionally at \(t=0\) and satisfy the side conditions. We consider only \({\varvec{\xi }}_{1}\) with non-negative elements; that these sum to one is enforced by the Lagrangian. Here, \(\alpha \) and the symmetric matrix \({\varvec{\varLambda }}\) are Lagrange multipliers. Since the maximum eigenvalue is simple, we have by Magnus (1985) that \({\mathcal {I}}_{\nu }\left( {\varvec{\xi }}_{t}|{{\mathbf {R}}}\right) \), hence \(\phi \left( t\right) \), is differentiable as well as convex, so that minimization of \(\phi \left( t\right) \) at \(t=0\) is equivalent to the condition \(\phi ^{\prime }\left( 0\right) \ge 0\), for all \({\varvec{\xi }}_{1}.\)

Denote by \({\mathbf {z}}_{I}\left( {\varvec{\xi }}_{0}|{{\mathbf {R}}} \right) \) the unit eigenvector belonging to the maximum eigenvalue of \( {\mathbf {U}}\left( {\varvec{\xi }}_{0}|{{\mathbf {R}}}\right) ={\mathbf {R}} ^{-1}{\mathbf {S}}\left( {\varvec{\xi }}_{0}\right) {\mathbf {R}}^{-1}\), so that \(\left\| {\mathbf {z}}_{I}\left( {\varvec{\xi }}_{0}|{{\mathbf {R}}} \right) \right\| =1\) and \({\mathcal {I}}_{\nu }\left( {\varvec{\xi }}_{0}| {{\mathbf {R}}}\right) ={\mathbf {z}}_{I}^{\prime }({\varvec{\xi }}_{0}| {{\mathbf {R}}})\left[ {\mathbf {R}}^{-1}{\mathbf {S}}\left( {\varvec{\xi }} _{0}\right) {\mathbf {R}}^{-1}\right] {\mathbf {z}}_{I}({\varvec{\xi }}_{0}| {{\mathbf {R}}})\). By Theorem 1 of Magnus (1985),

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}{\mathcal {I}}_{\nu }\left( {\varvec{\xi }}_{t}|{{\mathbf {R}} }\right) _{|_{t=0}}\\&\quad ={\mathbf {z}}_{I}^{\prime }({\varvec{\xi }}_{0}|{ {\mathbf {R}}})\left[ \frac{\mathrm{d}}{\mathrm{d}t}{\mathbf {R}}^{-1}{\mathbf {S}}\left( \varvec{ \xi }_{t}\right) {\mathbf {R}}^{-1}\right] _{|_{t=0}}{\mathbf {z}}_{I}(\varvec{ \xi }_{0}|{{\mathbf {R}}}), \end{aligned}$$

whence

$$\begin{aligned}&\frac{1}{2}\phi ^{\prime }\left( 0\right) ={\mathbf {z}}_{I}^{\prime }( {\varvec{\xi }}_{0}|{{\mathbf {R}}}){\mathbf {R}}^{-1}{\mathbf {Q}}^{\prime } {\mathbf {D}}\left( {\varvec{\xi }}_{0}\right) \left( {\mathbf {D}}\left( {\varvec{\xi }}_{1}\right) -{\mathbf {D}}\left( {\varvec{\xi }}_{0}\right) \right) {{\mathbf {Q}}}{{\mathbf {R}}}^{-1}{\mathbf {z}}_{I}({\varvec{\xi }}_{0}|{\mathbf { R}}) \nonumber \\&\quad -tr{\varvec{\varLambda }}\left( {\mathbf {Q}}^{\prime }\left( {\mathbf {D}}\left( {\varvec{\xi }}_{1}\right) -{\mathbf {D}}\left( {\varvec{\xi }}_{0}\right) \right) {\mathbf {Q}}\right) -2\alpha {\mathbf {1}}_{N}^{\prime }\left( {\varvec{\xi }}_{1}-{\varvec{\xi }}_{0}\right) \nonumber \\&=tr\left[ {\mathbf {D}}\left( {\varvec{\xi }}_{1}\right) -{\mathbf {D}}\left( {\varvec{\xi }}_{0}\right) \right] \left[ {{\mathbf {Q}}}{{\mathbf {R}}}^{-1}{\mathbf {z}}_{I}( {\varvec{\xi }}_{0}|{{\mathbf {R}}}){\mathbf {z}}_{I}^{\prime }( {\varvec{\xi }}_{0}|{{\mathbf {R}}}){\mathbf {R}}^{-1}{\mathbf {Q}}^{\prime } {\mathbf {D}}\left( {\varvec{\xi }}_{0}\right) -{\mathbf {Q}}{\varvec{\varLambda }}{\mathbf {Q}}^{\prime }\right] \nonumber \\&\quad -\alpha {\mathbf {1}}_{N}^{\prime }\left( {\varvec{\xi }}_{1}-\varvec{ \xi }_{0}\right) \nonumber \\&=\sum _{i=1}^{N}\left( \xi _{1,i}-\xi _{0,i}\right) \left\{ \begin{array}{c} \left( {{\mathbf {Q}}}{{\mathbf {R}}}^{-1}{\mathbf {z}}_{I}({\varvec{\xi }}_{0}|\mathbf {\mathbf {R }}){\mathbf {z}}_{I}^{\prime }({\varvec{\xi }}_{0}|{{\mathbf {R}}}) {\mathbf {R}}^{-1}{\mathbf {Q}}^{\prime }\right) _{ii}\xi _{0,i} \\ {-}\left( \left( {\mathbf {Q}}{\varvec{\varLambda }}{\mathbf {Q}}^{\prime }\right) _{ii}{+}\alpha \right) \end{array} \right\} . \end{aligned}$$
(33)

In order that \(\phi ^{\prime }\left( 0\right) \) be non-negative for all \(\xi _{1}\) it is necessary and sufficient that

$$\begin{aligned} \xi _{0,i}= & {} \frac{\left( \left( {\mathbf {Q}}{{\varvec{\varLambda }}}{\mathbf {Q}} ^{\prime }\right) _{ii}{+}\alpha \right) ^{+}}{\left( {{\mathbf {Q}}}{{\mathbf {R}}}^{-1}\mathbf {z }_{I}({\varvec{\xi }}_{0}|\mathbf {{\mathbf {R}}}){\mathbf {z}}_{I}^{\prime }( {\varvec{\xi }}_{0}|{{\mathbf {R}}}){\mathbf {R}}^{-1}{\mathbf {Q}}^{\prime }\right) _{ii}}\nonumber \\= & {} \frac{\left( {\mathbf {q}}_{i}^{\prime }{\varvec{\varLambda }} {\mathbf {q}}_{i}+\alpha \right) ^{+}}{\left( {\mathbf {q}}_{i}^{\prime } \varvec{\beta }\right) ^{2}}, \end{aligned}$$
(34)

where \(\varvec{\beta }={\mathbf {R}}^{-1}{\mathbf {z}}_{I}({\varvec{\xi }} _{0}|\mathbf {{\mathbf {R}}})\).

We verify the sufficiency of (34); the proof of necessity runs along the same lines. If (34) holds, then the set \(\left\{ i\left| {}\right. \xi _{0,i}>0\right\} \) coincides with the set \(\left\{ i\left| {}\right. {\mathbf {q}}_{i}^{\prime }{\varvec{\varLambda }}{\mathbf {q}}_{i}+\alpha >0\right\} \) and so (33) is

$$\begin{aligned}&\sum _{\left\{ \xi _{0,i}>0\right\} }\left( \xi _{1,i}-\xi _{0,i}\right) \left\{ \left( {\mathbf {q}}_{i}^{\prime }\varvec{\beta }\right) ^{2}\xi _{0,i}-\left( {\mathbf {q}}_{i}^{\prime }{\varvec{\varLambda }}{\mathbf {q}} _{i}+\alpha \right) ^{+}\right\} \\&\quad \quad +\sum _{\left\{ \xi _{0,i}=0\right\} }\left( -\xi _{1,i}\right) \left( {\mathbf {q}}_{i}^{\prime }{\varvec{\varLambda }} {\mathbf {q}}_{i}+\alpha \right) \\&\quad =0+\sum _{\left\{ \xi _{0,i}=0\right\} }\xi _{1,i}\left| {\mathbf {q}} _{i}^{\prime }{\varvec{\varLambda }}{\mathbf {q}}_{i}+\alpha \right| \ge 0, \end{aligned}$$

for non-negative \(\left\{ \xi _{1,i}\right\} \).

The multipliers \(\alpha \) and \({\varvec{\varLambda }}\) are to satisfy \( {\mathbf {1}}_{N}^{\prime }{\varvec{\xi }}_{0}=1\) and

$$\begin{aligned} {\mathbf {Q}}^{\prime }{\mathbf {D}}\left( {\varvec{\xi }}_{0}\right) {\mathbf {Q}}= \mathbf {R;} \end{aligned}$$
(35)

since (34) is overparameterized we can as well take \(\left\| \varvec{\beta }\right\| =1\). Then, \({\mathbf {R}}\) is varied so as to minimize \({\mathcal {I}}_{\nu }\left( {\varvec{\xi }}\right) =\left( 1-\nu \right) tr{\mathbf {R}}^{-1}\left( {\varvec{\xi }}\right) +\ \nu {\mathcal {I}} _{\nu }\left( {\varvec{\xi }}|{{\mathbf {R}}}\right) \). It is simpler however to define \({\varvec{\xi }}_{0}\) parametrically by (34) and to then choose the parameters to minimize (7a), subject to \(\mathbf {1 }_{N}^{\prime }{\varvec{\xi }}_{0}=1\) and \(\left\| \varvec{\beta } \right\| =1\), and with \({\mathbf {R}}\) defined by (35).

In this formulation it is clear that the solution for the D-criterion has the same form. \(\square \)

We give only the proof of Theorem 5, omitting the very similar proof of Theorem 6.

Proof of Theorem 5

Once (16) is established, the proof of Theorem 5 is completed by noting that, using (12) and then the definition of \( i^{*}\),

$$\begin{aligned} 0= & {} \sum _{i=1}^{N}\xi _{n,i}\left\{ \frac{{{\mathbf {T}}}_{ii}^{{\mathcal {I}} }\left( {\varvec{\xi }}_{n}\right) }{c\left( {\varvec{\xi }}_{n}; {\mathcal {I}}\right) }-1\right\} \le \max _{1\le i\le N}\left\{ \frac{{{\mathbf {T}}}_{ii}^{{\mathcal {I}}}\left( {\varvec{\xi }}_{n}\right) }{ c\left( {\varvec{\xi }}_{n};{\mathcal {I}}\right) }-1\right\} \nonumber \\= & {} \frac{{{\mathbf {T}}}_{i^{*}i^{*}}^{{\mathcal {I}}}\left( {\varvec{\xi }} _{n}\right) }{c\left( {\varvec{\xi }}_{n};{\mathcal {I}}\right) }-1. \end{aligned}$$
(36)

This implies (17). The inequality in (36) is strict, i.e. the maximum strictly exceeds the weighted average, unless \(\varvec{ \xi }_{n}\) places all mass on points satisfying (18).

The \(O\left( n^{-1}\right) \) term in (16) may be obtained from ( 30), by setting \(t=1/\left( n+1\right) \), \({\varvec{\xi }}_{0}= {\varvec{\xi }}_{n}\) and \({\varvec{\xi }}_{1}\) the one-point design \( \varDelta _{i}\) with all mass at \({\mathbf {x}}_{i}\) in the expansion

$$\begin{aligned}&{\mathcal {I}}_{\nu }\left( \left( 1-t\right) {\varvec{\xi }}_{0}+t {\varvec{\xi }}_{1}\right) ={\mathcal {I}}_{\nu }\left( {\varvec{\xi }} _{0}\right) \\&\quad +\left[ \frac{\mathrm{d}}{\mathrm{d}t}{\mathcal {I}}_{\nu }\left( {\varvec{\xi }} _{t}\right) _{|_{t=0}}\right] t+remainder. \end{aligned}$$

But a discussion of the order of the remainder requires us to take a longer, more detailed approach. We first require updated values of \({\mathbf {R}}_{n}= {\mathbf {R}}\left( {\varvec{\xi }}_{n}\right) \), \({\mathbf {U}}_{n}={\mathbf {U}} \left( {\varvec{\xi }}_{n}\right) \), and the maximum eigenvalue and corresponding eigenvector \(\lambda _{n}=\lambda \left( {\varvec{\xi }} _{n}\right) \) and \({\mathbf {z}}_{n}={\mathbf {z}}\left( {\varvec{\xi }} _{n}\right) \) of \({\mathbf {U}}_{n}\). From

$$\begin{aligned}&{\mathbf {D}}_{n+1}^{(i)}=\frac{n}{n+1}\left( {\mathbf {D}}_{n}+\frac{1}{n}\mathbf { e}_{i}{\mathbf {e}}_{i}^{\prime }\right) \text { and }{\mathbf {D}} _{n+1}^{(i)2}\nonumber \\&\quad =\left( \frac{n}{n+1}\right) ^{2}\left( {\mathbf {D}}_{n}^{2}+ \frac{2n_{i}+1}{n^{2}}{\mathbf {e}}_{i}{\mathbf {e}}_{i}^{\prime }\right) , \end{aligned}$$

we obtain

$$\begin{aligned}&{\mathbf {R}}_{n+1}^{(i)}=\frac{n}{n+1}\left( {\mathbf {R}}_{n}+\frac{1}{n}\mathbf { q}_{i}{\mathbf {q}}_{i}^{\prime }\right) \text { and }{\mathbf {S}} _{n+1}^{(i)}\nonumber \\&\quad =\left( \frac{n}{n+1}\right) ^{2}\left( {\mathbf {S}}_{n}+\frac{ 2n_{i}+1}{n^{2}}{\mathbf {q}}_{i}{\mathbf {q}}_{i}^{\prime }\right) . \end{aligned}$$
(37)

Then

$$\begin{aligned} \left[ {\mathbf {R}}_{n+1}^{(i)}\right] ^{-1}=\left( \frac{n+1}{n}\right) \left[ {\mathbf {R}}_{n}^{-1}-\frac{{\mathbf {R}}_{n}^{-1}{\mathbf {q}}_{i}{\mathbf {q}} _{i}^{\prime }{\mathbf {R}}_{n}^{-1}}{n+{\mathbf {q}}_{i}^{\prime }{\mathbf {R}} _{n}^{-1}{\mathbf {q}}_{i}}\right] , \end{aligned}$$

with

$$\begin{aligned} tr\left[ {\mathbf {R}}_{n+1}^{(i)}\right] ^{-1}=tr{\mathbf {R}}_{n}^{-1}+\frac{1}{n }\left[ tr{\mathbf {R}}_{n}^{-1}-{\mathbf {q}}_{i}^{\prime }{\mathbf {R}}_{n}^{-2} {\mathbf {q}}_{i}\right] +\frac{\gamma _{n,i}^{(1)}}{n^{2}}, \end{aligned}$$

for

$$\begin{aligned} \gamma _{n,i}^{(1)}=\frac{\left( {\mathbf {q}}_{i}^{\prime }{\mathbf {R}}_{n}^{-2} {\mathbf {q}}_{i}\right) \left( {\mathbf {q}}_{i}^{\prime }{\mathbf {R}}_{n}^{-1} {\mathbf {q}}_{i}-1\right) }{1+\frac{{\mathbf {q}}_{i}^{\prime }{\mathbf {R}}_{n}^{-1} {\mathbf {q}}_{i}}{n}}. \end{aligned}$$

Substituting (37) into \({\mathbf {U}}_{n+1}^{(i)}=\left[ {\mathbf {R}} _{n+1}^{(i)}\right] ^{-1}{\mathbf {S}}_{n+1}^{(i)}\left[ {\mathbf {R}}_{n+1}^{(i)} \right] ^{-1}\) yields, after a calculation,

$$\begin{aligned} {\mathbf {U}}_{n+1}^{(i)}={\mathbf {U}}_{n}-\frac{1}{n}{\mathbf {V}}_{n,i}-\frac{1}{ n^{2}}{\mathbf {W}}_{n,i}, \end{aligned}$$

for

$$\begin{aligned} {\mathbf {V}}_{n,i}= & {} {\mathbf {R}}_{n}^{-1}{\mathbf {q}}_{i}{\mathbf {q}}_{i}^{\prime } {\mathbf {U}}_{n}+{\mathbf {U}}_{n}{\mathbf {q}}_{i}{\mathbf {q}}_{i}^{\prime }{\mathbf {R}} _{n}^{-1}-2\xi _{n,i}{\mathbf {R}}_{n}^{-1}{\mathbf {q}}_{i}{\mathbf {q}}_{i}^{\prime }{\mathbf {R}}_{n}^{-1}, \\ {\mathbf {W}}_{n,i}= & {} -\left[ {\mathbf {R}}_{n}^{-1}{\mathbf {q}}_{i}{\mathbf {q}} _{i}^{\prime }{\mathbf {U}}_{n}\right. \\&\quad \left. +\, {\mathbf {U}}_{n}{\mathbf {q}}_{i}{\mathbf {q}} _{i}^{\prime }{\mathbf {R}}_{n}^{-1}\right] \left[ \frac{{\mathbf {q}}_{i}^{\prime }{\mathbf {R}}_{n}^{-1}{\mathbf {q}}_{i}}{1+\frac{{\mathbf {q}}_{i}^{\prime }\mathbf {R }_{n}^{-1}{\mathbf {q}}_{i}}{n}}\right] \\&\quad +\,\left[ {\mathbf {R}}_{n}^{-1}{\mathbf {q}}_{i}{\mathbf {q}}_{i}^{\prime }{\mathbf {R}} _{n}^{-1}\right] \\&\quad \left[ \frac{2\xi _{n,i}\left( 2{\mathbf {q}}_{i}^{\prime } {\mathbf {R}}_{n}^{-1}{\mathbf {q}}_{i}+\frac{\left( {\mathbf {q}}_{i}^{\prime } {\mathbf {R}}_{n}^{-1}{\mathbf {q}}_{i}\right) ^{2}}{n}\right) -\left( 1+{\mathbf {q}} _{i}^{\prime }{\mathbf {U}}_{n}{\mathbf {q}}_{i}\right) }{\left( 1+\frac{{\mathbf {q}} _{i}^{\prime }{\mathbf {R}}_{n}^{-1}{\mathbf {q}}_{i}}{n}\right) ^{2}}\right] . \end{aligned}$$

Substituting these expressions into \({\mathcal {I}}_{\nu }\left( \varvec{ \xi }_{n+1}^{(i)}\right) =\left( 1-\nu \right) tr\left[ {\mathbf {R}} _{n+1}^{(i)}\right] ^{-1}+\nu ch_{\max }\left[ {\mathbf {U}}_{n+1}^{(i)}\right] \) gives

$$\begin{aligned}&{\mathcal {I}}_{\nu }\left( {\varvec{\xi }}_{n+1}^{(i)}\right) =\left( 1-\nu \right) \left\{ tr{\mathbf {R}}_{n}^{-1}+\frac{1}{n}\left[ tr{\mathbf {R}} _{n}^{-1}-{\mathbf {q}}_{i}^{\prime }{\mathbf {R}}_{n}^{-2}{\mathbf {q}}_{i}\right] + \frac{\gamma _{n,i}^{(1)}}{n^{2}}\right\} \nonumber \\&\qquad +\,\nu \rho _{i}\left( \frac{1}{n} \right) , \end{aligned}$$
(38)

where we define \(\rho _{i}\left( t\right) =ch_{\max }{\mathbf {A}}_{i}\left( t\right) \) for \({\mathbf {A}}_{i}\left( t\right) ={\mathbf {U}}_{n}-t{\mathbf {V}} _{n,i}-t^{2}{\mathbf {W}}_{n,i}\). Now let \({\mathbf {z}}_{(i)}\left( t\right) \) denote the eigenvector of unit norm corresponding to \(\rho _{i}\left( t\right) \), set \({\mathbf {Y}}_{i}\left( t\right) =\phi \left( t\right) \mathbf { I}_{p}-{\mathbf {A}}_{i}\left( t\right) \) and denote by \({\mathbf {Y}}_{i}^{+}\) the Moore-Penrose inverse. Then, for some \(t_{n}\in \left[ 0,1\right] \), \( \rho _{i}\left( \frac{1}{n}\right) =\rho _{i}\left( 0\right) +\rho _{i}^{\prime }\left( 0\right) /n+\frac{1}{2}\rho _{i}^{\prime \prime }\left( \frac{t_{n}}{n}\right) /n^{2}\). Using Theorems 1 and 4 of Magnus (1985) to evaluate the derivatives, this becomes

$$\begin{aligned} \rho _{i}\left( \frac{1}{n}\right) =\lambda _{n}-\frac{1}{n}{\mathbf {z}} _{n}^{\prime }{\mathbf {V}}_{n,i}{\mathbf {z}}_{n}+\frac{\gamma _{n,i}^{(2)}}{n^{2} }, \end{aligned}$$
(39)

where

$$\begin{aligned}&\gamma _{n,i}^{(2)}=\frac{1}{2}\rho _{i}^{\prime \prime }\left( \frac{t_{n} }{n}\right) \nonumber \\&\quad ={\mathbf {z}}_{(i)}^{\prime }\left( \frac{t_{n}}{n}\right) \left\{ \left( {\mathbf {V}}_{n,i}+2\frac{t_{n}}{n}{\mathbf {W}}_{n,i}\right) {\mathbf {Y}} _{i}^{+}\left( \frac{t_{n}}{n}\right) \left( {\mathbf {V}}_{n,i}+2\frac{t_{n}}{n }{\mathbf {W}}_{n,i}\right) \right\} \nonumber \\&\qquad \times \, {\mathbf {z}}_{(i)}\left( \frac{t_{n}}{n} \right) \nonumber \\&\qquad -\, {\mathbf {z}}_{(i)}^{\prime }\left( \frac{t_{n}}{n}\right) {\mathbf {W}}_{n,i} {\mathbf {z}}_{(i)}\left( \frac{t_{n}}{n}\right) . \end{aligned}$$
(40)

Substituting (40) and (39) into (38) gives ( 16), with

$$\begin{aligned} \beta _{n,i}^{I}=\left( 1-\nu \right) \gamma _{n,i}^{(1)}+\nu \gamma _{n,i}^{(2)}. \end{aligned}$$

Under the assumption on the eigenvalues of \({\mathbf {R}}_{n}\) the elements of \( {\mathbf {R}}_{n}^{-1}\), \({\mathbf {V}}_{n,i}\), and \({\mathbf {W}}_{n,i}\) remain bounded, hence so does \(\beta _{n,i}^{I}\). \(\square \)

Proof of Theorem 7

(i) Write the statements of Theorems 5 and 6 as

$$\begin{aligned} l_{n+1}=l_{n}-\frac{\alpha _{n}}{n}+\frac{\beta _{n}}{n^{2}}, \end{aligned}$$
(41)

where

$$\begin{aligned} l_{n}= & {} {\mathcal {L}}_{\nu }\left( {\varvec{\xi }}_{n}^{*}\right) , \\ \alpha _{n}= & {} T_{i^{*}i^{*}}^{{\mathcal {L}}}\left( {\varvec{\xi }} _{n}^{*}\right) -c\left( {\varvec{\xi }}_{n}^{*}\right) \ge 0, \\ \beta _{n}= & {} n^{2}\left( l_{n+1}-l_{n}+\frac{\alpha _{n}}{n}\right) . \end{aligned}$$

Summing (41) over \(M_{1}\le n\le M_{2}-1\) and defining a positive, increasing sequence \(\left\{ a_{M_{2}}\right\} _{M_{2}>M_{1}}\) by \( a_{M_{2}}=\sum _{M_{1}}^{M_{2}-1}\left( \alpha _{n}/n\right) \) gives \( l_{M_{1}}-l_{M_{2}}=a_{M_{2}}-\sum _{M_{1}}^{M_{2}-1}\left( \beta _{n}/n^{2}\right) \). Since \(\sum _{n}n^{-2}\) is convergent and \(\left\{ \beta _{n}\right\} \) is bounded, for given \(\varepsilon >0\), we can then find \( M_{0} \) such that for \(M_{2}>M_{1}>M_{0}\),

$$\begin{aligned} \left| l_{M_{1}}-l_{M_{2}}-a_{M_{2}}\right| =\left| \sum _{M_{1}}^{M_{2}-1}\frac{\beta _{n}}{n^{2}}\right| <\frac{\varepsilon }{2}. \end{aligned}$$
(42)

This implies first that \(l_{M_{2}}\in \left( 0,l_{M_{1}}+\varepsilon /2\right) \), so that \(\left\{ l_{n}\right\} \) is bounded, and then that \( a_{M_{2}}\in \left( 0,l_{M_{1}}-l_{M_{2}}+\varepsilon /2\right) \), so that \( \left\{ a_{M_{2}}\right\} _{M_{2}>M_{1}}\) is bounded as well as increasing, hence convergent, hence Cauchy. So there is \(M_{0}^{\prime }\) such that for \( M_{2}>M_{1}>M_{0}^{\prime }\) we have

$$\begin{aligned} 0<a_{M_{2}}<\frac{\varepsilon }{2}. \end{aligned}$$
(43)

This together with (42) gives \(\left| l_{M_{1}}-l_{M_{2}}\right| <\varepsilon \) for \(M_{2}>M_{1}>\max \left( M_{0},M_{0}^{\prime }\right) \), i.e. \(\left\{ l_{n}\right\} \) is Cauchy, hence convergent.(ii) From (43) \(\sum _{n}\left( \alpha _{n}/n\right) \) converges even though \(\sum _{n}n^{-1}\) diverges; hence for every \( \varepsilon >0\), there must be terms \(\alpha _{n}<\varepsilon \). Thus, there is a subsequence \(\left\{ \alpha _{n_{j}}\right\} _{j=1}^{\infty }\) with \( \alpha _{n_{j}}\downarrow 0\). This implies (21):

$$\begin{aligned} \delta \left( {\varvec{\xi }}_{n_{j}}^{*}\right)= & {} \frac{ \max _{i}T_{ii}^{{\mathcal {L}}}\left( {\varvec{\xi }}_{n_{j}}^{*}\right) }{\sum _{\xi _{n_{j},i}^{*}>0}\xi _{n,i}^{*}T_{ii}^{{\mathcal {L}} }\left( {\varvec{\xi }}_{n_{j}}^{*}\right) }-1\nonumber \\= & {} \frac{\alpha _{n_{j}}+c\left( {\varvec{\xi }}_{n_{j}}^{*};{\mathcal {L}}\right) }{ c\left( {\varvec{\xi }}_{n_{j}}^{*};{\mathcal {L}}\right) }-1\downarrow 0. \end{aligned}$$

\(\square \)

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Wiens, D.P. I-robust and D-robust designs on a finite design space. Stat Comput 28, 241–258 (2018). https://doi.org/10.1007/s11222-017-9728-8

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