The stochastic gradient descent (SGD) algorithm has been widely used to optimize deep Cox neural network (Cox-NN) by updating model parameters using mini-batches of data. We show that SGD aims to optimize the average of mini-batch partial-likelihood, which is different from the standard partial-likelihood. This distinction requires developing new statistical properties for the global optimizer, namely, the mini-batch maximum partial-likelihood estimator (mb-MPLE). We establish that mb-MPLE for Cox-NN is consistent and achieves the optimal minimax convergence rate up to a polylogarithmic factor. For Cox regression with linear covariate effects, we further show that mb-MPLE is n -consistent and asymptotically normal with asymptotic variance approaching the information lower bound as batch size increases, which is confirmed by simulation studies. Additionally, we offer practical guidance on using SGD, supported by theoretical analysis and numerical evidence. For Cox-NN, we demonstrate that the ratio of the learning rate to the batch size is critical in SGD dynamics, offering insight into hyperparameter tuning. For Cox regression, we characterize the iterative convergence of SGD, ensuring that the global optimizer, mb-MPLE, can be approximated with sufficiently many iterations. Finally, we demonstrate the effectiveness of mb-MPLE in a large-scale real-world application where the standard MPLE is intractable.
In the general setting of independent data with possibly very different distributions, extreme value estimators inevitably target the tail of the average distribution function. We consider all possible cases, that is, the extreme value index of the average distribution can be negative, zero, or positive, and we present novel asymptotic theory for the moment estimator. Our results require a different and much more challenging proof than those for the power-law case and are based on a uniform central limit theorem for the underlying weighted tail empirical process. We find that, due to the heterogeneity of the data, the asymptotic variance of the moment estimator can be much smaller than that in the iid case. We also unravel the improved performance of high quantile and endpoint estimators in this setup. In case of a heavy tail, we ameliorate the Hill estimator by taking an optimal combination of the Hill and the moment estimator. Simulations show the good finite-sample behavior of our limit results. Finally we present applications to the maximum lifespan of monozygotic twins, the ultimate 200m running world records, and to the tail heaviness of energies of earthquakes around the globe. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.
The organization of human brain subnetworks is fundamental to understanding cognition and neuropsychiatric health. Existing approaches predominantly construct subnetworks by clustering brain regions according to measured imaging phenotypes or functional correlation. Although successful, such phenotype-based parcellations reflect composite effects of genetics, environment, lifestyle, and measurement noise, thereby limiting biological interpretability and obscuring subnetworks attributable to specific mechanisms. Accumulating evidence suggests that brain connectivity is substantially heritable and genetic influences are regionally heterogeneous and aligned with functional architecture. Motivated by these findings, we propose a novel statistical framework to construct genetically informed brain subnetworks by disentangling genetic effect from other sources of variation. The framework integrates two key components: a high-dimensional multi-task learning model that decomposes composite effects, and a group-wise mixture structure on gene-specific regional effects to identify latent clustering patterns induced by individual genes. This formulation enables direct modeling of regional genetic influences rather than relying on aggregate heritability measures, yielding subnetworks with shared molecular mechanisms. We further develop an iterative algorithm for model fitting, and establish theoretical guarantees for its efficiency, supported by simulation results. This framework is general and can be extended to derive subnetworks induced by other factors of interest. We implement the method on a dataset from the UK Biobank (UKB) for discovery, and validate the results on the Human Connectome Project Young Adult (HCP-YA) dataset. Our results reveal intrinsic, heritable brain atlases that complement conventional phenotype-based parcellations and provides new insight into the interplay among genes, brain organization, and neuropsychiatric disorders. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.
