On the irregular behavior of LS estimators for Asymptotically singular designs ## AbstractOptimum 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 $\xi^*$.
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