To provide a solid analytic foundation for the module approach to conditional risk measures, our purpose is to establish a complete random convex analysis over random locally convex modules by simultaneously considering the two kinds of topologies (namely the (ε, λ)-topology and the locally L 0 -convex topology). This paper is focused on the part of separation and Fenchel-Moreau duality in random locally convex modules. The key point of this paper is to give the precise relation between random conjugate spaces of a random locally convex module under the two kinds of topologies, which enables us to not only give a thorough treatment of separation between a point and a closed L 0 -convex subset but also establish the complete Fenchel-Moreau duality theorems in random locally convex modules under the two kinds of topologies.
Combining respective advantages of the (ε, λ)-topology and the locally L0-convex topology we first prove that every complete random normed module is random subreflexive under the (ε, λ)-topology. Further, we prove that every complete random normed module with the countable concatenation property is also random subreflexive under the locally L0-convex topology, at the same time we also provide a counterexample which shows that it is necessary to require the random normed module to have the countable concatenation property.
When designing an experiment, it is important to choose a design that is optimal under model uncertainty. The general minimum lower-order confounding (GMC) criterion can be used to control aliasing among lower-order factorial effects. A characterization of GMC via complementary sets was considered in Zhang and Mukerjee (2009a); however, the problem of constructing GMC designs is only partially solved. We provide a solution for two-level factorial designs with n factors and N = 2 n−m runs subject to a restriction on (n, N): 5N/16 + 1 ≤ n ≤ N − 1. The construction is quite simple: every GMC design, up to isomorphism, consists of the last n columns of the saturated 2 (N −1)−(N −1−n+m) design with Yates order. In addition, we prove that GMC designs differ from minimum aberration designs when (n, N) satisfies either of the following conditions: (i) 5N/16 + 1 ≤ n ≤ N/2 − 4, or (ii) n ≥ N/2, 4 ≤ n + 2 r − N ≤ 2 r−1 − 4 with r ≥ 4.
It has been well acknowledged that methods for secondary trait (ST) association analyses under a case-control design (ST$_{\text{CC}}$) should carefully consider the sampling process to avoid biased risk estimates. A similar situation also exists in the extreme phenotype sequencing (EPS) designs, which is to select subjects with extreme values of continuous primary phenotype for sequencing. EPS designs are commonly used in modern epidemiological and clinical studies such as the well-known National Heart, Lung, and Blood Institute Exome Sequencing Project. Although naïve generalized regression or ST$_{\text{CC}}$ method could be applied, their validity is questionable due to difference in statistical designs. Herein, we propose a general prospective likelihood framework to perform association testing for binary and continuous STs under EPS designs (STEPS), which can also incorporate covariates and interaction terms. We provide a computationally efficient and robust algorithm to obtain the maximum likelihood estimates. We also present two empirical mathematical formulas for power/sample size calculations to facilitate planning of binary/continuous STs association analyses under EPS designs. Extensive simulations and application to a genome-wide association study of benign ethnic neutropenia under an EPS design demonstrate the superiority of STEPS over all its alternatives above.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.