To fast approximate the maximum likelihood estimator with massive data, Wang et al. (JASA, 2017) proposed an optimal subsampling method under the A-optimality criterion (OSMAC) for in logistic regression. This paper extends the scope of the OS-MAC framework to include generalized linear models with canonical link functions. The consistency and asymptotic normality of the estimator from a general subsampling algorithm are established, and optimal subsampling probabilities under the Aand L-optimality criteria are derived. Furthermore, using Frobenius norm matrix concentration inequality, finite sample properties of the subsample estimator based on optimal subsampling probabilities are derived. Since the optimal subsampling probabilities depend on the full data estimate, an adaptive two-step algorithm is developed. Asymptotic normality and optimality of the estimator from this adaptive algorithm are established. The proposed methods are illustrated and evaluated through numerical experiments on simulated and real datasets.

New types of designs called nested space-filling designs have been proposed for conducting multiple computer experiments with different levels of accuracy. In this article, we develop several approaches to constructing such designs. The development of these methods also leads to the introduction of several new discrete mathematics concepts, including nested orthogonal arrays and nested difference matrices.Comment: Published in at http://dx.doi.org/10.1214/09-AOS690 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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The interference model has been widely used and studied in block experiments where the treatment for a particular plot has effects on its neighbor plots. In this paper, we study optimal circular designs for the proportional interference model, in which the neighbor effects of a treatment are proportional to its direct effect. Kiefer's equivalence theorems for estimating both the direct and total treatment effects are developed with respect to the criteria of A, D, E and T. Parallel studies are carried out for the undirectional model, where the neighbor effects do not depend on whether they are from the left or right. Moreover, the connection between optimal designs for the directional and undiretional models is built. Importantly, one can easily develop a computer program for finding optimal designs based on these theorems.1. Introduction. In many agricultural experiments, the treatment assigned to a particular plot could also have effects on its neighbor plots. This is well recognized in literature. See Draper and Guttman (1980), Kempton (1982), Besag and Kempton (1986), Langton (1990), Gill (1993 and Goldringer, Brabant and Kempton (1994), for examples. To adjust the biases caused by these neighbor effects, the interference model is widely adopted. In a block design with n blocks of size k and t treatments, the response, y dij , observed from the jth plot of block i is decomposed into the following items: