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 [theta], or of a function of interest h([theta]), may be impossible with the singular design [xi]*. Depending on how [xi]* is approached by the empirical measure [xi]n of the design points, with n the number of observations, consistency is achieved but the speed of convergence may depend on [xi]n and on the value of [theta]. Even in situations where convergence is in and the asymptotic distribution of the estimator of [theta] or h([theta]) is normal, the asymptotic variance may still differ from that obtained from [xi]*.