On the irregular behavior of LS estimators for asymptotically singular designs

Optimum design theory sometimes yields singular designs. An example with a linear regression model often mentioned in the literature is used to illustrate the difficulties induced by such designs. The estimation of the model parameters θ, or of a function of interest h(θ), may be impossible with the singular design ξ*. Depending on how ξ* is approached by the empirical measure ξn of the design points, with n the number of observations, consistency is achieved but the speed of convergence may depend on ξn and on the value of θ. Even in situations where convergence is in 1/n and the asymptotic distribution of the estimator of θ or h(θ) is normal, the asymptotic variance may still differ from that obtained from ξ*.