A delimitation of the support of optimal designs for Kiefer's ${\varphi }_p$-class of criteria

Abstract

The paper extends the result of Harman and Pronzato [Stat. & Prob. Lett., 77:90--94, 2007], which corresponds to $p=0$, to all strictly concave criteria in Kiefer's ${\varphi }_p$-class. Let $\xi$ be any design on a compact set $X\subset {\mathbb{R}}^m$ with a nonsingular information matrix $\Mb(ξ)$, and let $\delta$ be the maximum of the directional derivative $F_{{\varphi }_p}(\xi ,x)$ over all $x\in X$. We show that any support point $x_{\ast }$ of a ${\varphi }_p$-optimal design satisfies the inequality $F_{ϕ_p}(ξ,x_*) \geq h_p[\Mb(ξ),δ]$, where the bound $h_p[\Mb(ξ),δ]$ is easily computed: it requires the determination of the unique root of a simple univariate equation (polynomial when $p$ is integer) in a given interval. The construction can be used to accelerate algorithms for ${\varphi }_p$-optimal design and is illustrated on an example with $A$-optimal design.