A delimitation of the support of optimal designs for Kiefer's ϕₚ-class of criteria

Original · EN

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 ϕₚ-class. Let ξ be any design on a compact set Xᵐ with a nonsingular information matrix (ξ), and let δ be the maximum of the directional derivative Fϕₚ(ξ,x) over all x∈ X. We show that any support point x* of a ϕₚ-optimal design satisfies the inequality Fϕₚ(ξ,x*) ≥ hₚ[(ξ),δ], where the bound hₚ[(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 ϕₚ-optimal design and is illustrated on an example with A-optimal design.