Local optimality of replications from a minimal XX-optimal design in regression: A sufficient and quasi-necessary condition

This paper aims at further studying the properties of XX- and X′X′-optimal design criteria, which are based on the minimisation of the expected volumes of exact parameter confidence regions in nonlinear regression. The definition and main characteristics of these criteria have recently been published by the authors of the present paper in another journal. This paper is devoted to the analysis of the frequent optimality of designs made of replications of minimal XX- and X′X′-optimal designs of size equal to the number of model parameters and heteroscedastic variance error structures. When it can be anticipated, this property makes it possible to drastically reduce the optimisation problem dimension and, subsequently, computation time. Using a method similar to that of a previous study of the same property for the DD-optimality criterion by one of the authors, a sufficient condition for strong local optimality of such replicated designs is given. A necessary condition is obtained by slightly weakening this sufficient condition which, for most practical purposes, can then be considered as necessary and sufficient. Several case studies show the practicability of this condition.