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On the irregular behavior of LS estimators for asymptotically singular designs

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 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∗.
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Andrej Pazman, Luc Pronzato. On the irregular behavior of LS estimators for asymptotically singular designs. Statistics and Probability Letters, Elsevier, 2006, 76, pp.1089-1096. ⟨10.1016/j.spl.2005.12.010⟩. ⟨hal-00416062⟩

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