From the authors’ abstract: “The Lasso approach is widely adopted for screening and estimating active effects in sparse linear models with quantitative factors. Many design schemes have been proposed based on different criteria to make the Lasso estimator more accurate. This article applies \(\Phi_l\)-optimality to the asymptotic covariance matrix of the Lasso estimator.”
The \(\Phi_l\) optimality considered in this paper is defined through the eigenvalues of the asymptotic covariance matrix \(M\) of the Lasso estimator. Thus, \(\Phi_l(M)\) is defined as \(\Phi_l (M) = ({\mathrm{tr}} M^l )^{1/l}\). When the parameter \(l\) ranges over \([0, +\infty]\), one obtains different notions of optimality, namely: \begin{align*} \Phi_0 (M) = \det(M) \qquad & \text{\(D\)-optimality} \\ \Phi_1 (M) = \operatorname{tr}(M) \qquad &\text{\(A\)-optimality} \\ \Phi_\infty(M) = \text{max eigenvalue of \(M\)} \qquad & \text{\(E\)-optimality}. \end{align*} A simulation study with several scenarios and an example based on real-world data are described in order to show the performance of the estimation procedure under the proposed design scheme.