On the irregular behavior of LS estimators for Asymptotically singular designs
Andrej Pazman and Luc Pronzato
Statistics and Probability Letters 2005.

Abstract

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 $\mt$, or of a function of interest $h(\mt)$, 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 $\mt$. Even in situations where convergence is in $1/\sqrt{n}$ and the asymptotic distribution of the estimator of $\mt$ or $h(\mt)$ is normal, the asymptotic variance may still differ from that obtained from $\xi^*$.