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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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