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Robustness of $$A$$-optimal designs. (English) Zbl 1145.62053
Summary: Suppose that $$Y=(Y_i)$$ is a normal random vector with mean $$Xb$$ and covariance $$\sigma^2l_n$$, where $$b$$ is a $$p$$-dimensional vector $$(b_j)$$, $$X=(X_{ij})$$ is an $$n\times p$$ matrix. $$A$$-optimal designs $$X$$ are chosen from the traditional set $$\mathcal D$$ of $$A$$-optimal designs for $$\rho=0$$ such that $$X$$ is still $$A$$-optimal in $$\mathcal D$$ when the components $$Y_i$$ are dependent, i.e., for $$i\neq i'$$, the covariance of $$Y_i$$, $$Y_{i'}$$ is $$\rho$$ with $$\rho\neq 0$$. Such designs depend on the sign of $$\rho$$. The general results are applied to $$X=(X_{ij})$$, where $$X_{ij}\in [-1,1]$$; this corresponds to a factorial design with $$-1,1$$ representing a low level or high level, respectively, or corresponds to a weighing design with $$-1,1$$ representing an object $$j$$ with weight $$b_j$$ being weighed on the left and right of a chemical balance, respectively.

##### MSC:
 62K05 Optimal statistical designs 62K15 Factorial statistical designs 62K25 Robust parameter designs 62H12 Estimation in multivariate analysis
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