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Optimal designs for multiple-regression models are determined. We consider a general class of non-linear models including proportional hazards models with different censoring schemes, the Poisson and the negative binomial model. For these models we provide a complete characterization of c-optimal designs for all vectors c in the case of a single covariate. For multiple regression with an arbitrary number of covariates, c-optimal designs for certain vectors c are derived analytically. Using some general results on the structure of optimal designs for multiple regression, we determine L- and ϕk -optimal designs for models with an arbitrary number of covariates.

We propose a class of subspace ascent methods for computing optimal approximate designs that covers existing algorithms as well as new and more efficient ones. Within this class of methods, we construct a simple, randomized exchange algorithm (REX). Numerical comparisons suggest that the performance of REX is comparable or superior to that of state-of-the-art methods across a broad range of problem structures and sizes. We focus on the most commonly used criterion of D-optimality, which also has applications beyond experimental design, such as the construction of the minimum-volume ellipsoid containing a given set of data points. For D-optimality, we prove that the proposed algorithm converges to the optimum. We also provide formulas for the optimal exchange of weights in the case of the criterion of A-optimality, which enable one to use REX and some other algorithms for computing A-optimal and I-optimal designs. Supplementary materials for this article are available online.

We consider screening experiments where an investigator wishes to study many factors using fewer observations. Our focus is on experiments with two-level factors and a main effects model with intercept. Since the number of parameters is larger than the number of observations, traditional methods of inference and design are unavailable. In 1959 , Box suggested the use of supersaturated designs and in 1962 , Booth and Cox introduced measures for efficiency of these designs including E(s 2 ), which is the average of squares of the off-diagonal entries of the information matrix, ignoring the intercept. For a design to be E(s 2 )-optimal, the main effect of every factor must be orthogonal to the intercept (factors are balanced), and among all designs that satisfy this condition, it should minimize E(s 2 ). This is a natural approach since it identifies the most nearly orthogonal design, and orthogonal designs enjoy many desirable properties including efficient parameter estimation. Factor balance in an E(s 2 )-optimal design has the consequence that the intercept is the most precisely estimated parameter. We introduce and study UE(s 2 )-optimality, which is essentially the same as E(s 2 )-optimality, except that we do not insist on factor balance. We also provide a method of construction. We introduce a second criterion from a traditional design optimality theory viewpoint. We use minimization of bias as our estimation criterion, and minimization of the variance of the minimum bias estimator as the design optimality criterion. Using D-optimality as the specific design optimality criterion, we introduce D-optimal supersaturated designs. We show that D-optimal supersaturated designs can be constructed from D-optimal chemical balance weighing designs obtained by Galil and Kiefer ( 1980 , 1982 ), Cheng ( 1980 ) and other authors. It turns out that, except when the number of observations and the number of factors are in a certain range, an UE(s 2 )-optimal design is also a D-optimal supersaturated design. Moreover, these designs have an interesting connection to Bayes optimal designs. When the prior variance is large enough, a D-optimal supersaturated design is Bayes D-optimal and when the prior variance is small enough, an UE(s 2 )-optimal design is Bayes D-optimal. While E(s 2 )-optimal designs yield precise intercept estimates, our study indicates that UE(s 2 )-optimal designs generally produce more efficient estimates for the main effects of the factors. Based on theoretical properties and the study of examples, we recommend UE(s 2 )-optimal designs for screening experiments.

Global vegetation and land-surface models embody interdisciplinary scientific understanding of the behaviour of plants and ecosystems, and are indispensable to project the impacts of environmental change on vegetation and the interactions between vegetation and climate. However, systematic errors and persistently large differences among carbon and water cycle projections by different models highlight the limitations of current process formulations. In this review, focusing on core plant functions in the terrestrial carbon and water cycles, we show how unifying hypotheses derived from eco-evolutionary optimality (EEO) principles can provide novel, parameter-sparse representations of plant and vegetation processes. We present case studies that demonstrate how EEO generates parsimonious representations of core, leaf-level processes that are individually testable and supported by evidence. EEO approaches to photosynthesis and primary production, dark respiration and stomatal behaviour are ripe for implementation in global models. EEO approaches to other important traits, including the leaf economics spectrum and applications of EEO at the community level are active research areas. Independently tested modules emerging from EEO studies could profitably be integrated into modelling frameworks that account for the multiple time scales on which plants and plant communities adjust to environmental change.

This paper treats the problem of minimizing a general continuously differentiable function subject to sparsity constraints. We present and analyze several different optimality criteria which are based on the notions of stationarity and coordinatewise optimality. These conditions are then used to derive three numerical algorithms aimed at finding points satisfying the resulting optimality criteria: the iterative hard thresholding method and the greedy and partial sparse-simplex methods. The first algorithm is essentially a gradient projection method, while the remaining two algorithms are of a coordinate descent type. The theoretical convergence of these techniques and their relations to the derived optimality conditions are studied. The algorithms and results are illustrated by several numerical examples. [PUBLICATION ABSTRACT]

Response surface experiments often involve only quantitative factors, and the response is fit using a full quadratic model in these factors. The term response surface implies that interest in these studies is more on prediction than parameter estimation because the points on the fitted surface are predicted responses. When computing optimal designs for response surface experiments, it therefore makes sense to focus attention on the predictive capability of the designs. However, the most popular criterion for creating optimal experimental designs is the D-optimality criterion, which aims to minimize the variance of the factor effect estimates in an omnibus sense. Because I-optimal designs minimize the average variance of prediction over the region of experimentation, their focus is clearly on prediction. Therefore, the I-optimality criterion seems to be a more appropriate one than the D-optimality criterion for generating response surface designs. Here we introduce I-optimal design of split-plot response surface experiments. We show through several examples that I-optimal split-plot designs provide substantial benefits in terms of improved prediction compared with D-optimal split-plot designs, while also performing very well in terms of the precision of the factor effect estimates.

A common goal in statistics and machine learning is to learn models that can perform well against distributional shifts, such as latent heterogeneous subpopulations, unknown covariate shifts or unmodeled temporal effects. We develop and analyze a distributionally robust stochastic optimization (DRO) framework that learns a model providing good performance against perturbations to the data-generating distribution. We give a convex formulation for the problem, providing several convergence guarantees. We prove finitesample minimax upper and lower bounds, showing that distributional robustness sometimes comes at a cost in convergence rates. We give limit theorems for the learned parameters, where we fully specify the limiting distribution so that confidence intervals can be computed. On real tasks including generalizing to unknown subpopulations, fine-grained recognition and providing good tail performance, the distributionally robust approach often exhibits improved performance.

We construct optimal designs for group testing experiments where the goal is to estimate the prevalence of a trait by using a test with uncertain sensitivity and specificity. Using optimal design theory for approximate designs, we show that the most efficient design for simultaneously estimating the prevalence, sensitivity and specificity requires three different group sizes with equal frequencies. However, if estimating prevalence as accurately as possible is the only focus, the optimal strategy is to have three group sizes with unequal frequencies. On the basis of a chlamydia study in the USA we compare performances of competing designs and provide insights into how the unknown sensitivity and specificity of the test affect the performance of the prevalence estimator. We demonstrate that the locally D- and Ds-optimal designs proposed have high efficiencies even when the prespecified values of the parameters are moderately misspecified.

In mixture experiments, the factors under study are proportions of the ingredients of a mixture. The special nature of the factors necessitates specific types of regression models, and specific types of experimental designs. Although mixture experiments usually are intended to predict the response(s) for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients, little research has been done concerning their I-optimal design. This is surprising given that I-optimal designs minimize the average variance of prediction and, therefore, seem more appropriate for mixture experiments than the commonly used D-optimal designs, which focus on a precise model estimation rather than precise predictions. In this article, we provide the first detailed overview of the literature on the I-optimal design of mixture experiments and identify several contradictions. For the second-order and the special cubic model, we present continuous I-optimal designs and contrast them with the published results. We also study exact I-optimal designs, and compare them in detail to continuous I-optimal designs and to D-optimal designs. One striking result of our work is that the performance of D-optimal designs in terms of the I-optimality criterion very strongly depends on which of the D-optimal designs is considered. Supplemental materials for this article are available online.

To balance robustness of quantile regression and effectiveness of expectile regression, we consider Lp-quantile regression models with large-scale data and develop a unified optimal subsampling method to downsize the data volume and reduce computational burden. For low-dimensional Lp-quantile regression models, two optimal subsampling probabilities based on the A- and L-optimality criteria are firstly proposed. For the preconceived low-dimensional parameter in high-dimensional Lp-quantile regression models, a novel optimal subsampling decorrelated score function is proposed to mitigate the effect from nuisance parameter estimation and then two optimal decorrelated score subsampling probabilities are provided. The asymptotic properties of two optimal subsample estimators are established. The finite-sample performance of the proposed estimators is studied through simulations, and an application to Beijing Air Quality Dataset is also presented.