Toptimal designs for multifactor polynomial regression models via a semidefinite relaxation method. We consider Toptimal experiment design problems for discriminating multifactor polynomial regression models where the design space is defined by polynomial inequalities and the regression parameters are constrained to given convex sets. Our proposed optimality criterion is formulated as a convex optimization problem with a moment cone constraint. When the regression models have one factor, an exact semidefinite representation of the moment cone constraint can be applied to obtain an equivalent semidefinite program. When there are two or more factors in the models, we apply a moment relaxation technique and approximate the moment cone constraint by a hierarchy of semidefiniterepresentable outer approximations. When the relaxation hierarchy converges, an optimal discrimination design can be recovered from the optimal moment matrix, and its optimality is confirmed by an equivalence theorem. The methodology is illustrated with several examples.

Selection Consistency of LassoBased Procedures for Misspecified HighDimensional Binary Model and Random Regressors We consider selection of random predictors for a highdimensional regression problem with a binary response for a general loss function. An important special case is when the binary model is semiparametric and the response function is misspecified under a parametric model fit. When the true response coincides with a postulated parametric response for a certain value of parameter, we obtain a common framework for parametric inference. Both cases of correct specification and misspecification are covered in this contribution. Variable selection for such a scenario aims at recovering the support of the minimizer of the associated risk with large probability. We propose a twostep selection ScreeningSelection (SS) procedure which consists of screening and ordering predictors by Lasso method and then selecting the subset of predictors which minimizes the Generalized Information Criterion for the corresponding nested family of models. We prove consistency of the proposed selection method under conditions that allow for a much larger number of predictors than the number of observations. For the semiparametric case when distribution of random predictors satisfies linear regressions condition, the true and the estimated parameters are collinear and their common support can be consistently identified. This partly explains robustness of selection procedures to the response function misspecification.