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This article introduces an Ordinary Differential Equation (ODE) notion for survival analysis. The ODE notion not only provides a unified modeling framework, but more importantly, also enables the development of a widely applicable, scalable, and easy-to-implement procedure for estimation and inference. Specifically, the ODE modeling framework unifies many existing survival models, such as the proportional hazards model, the linear transformation model, the accelerated failure time model, and the time-varying coefficient model as special cases. The generality of the proposed framework serves as the foundation of a widely applicable estimation procedure. As an illustrative example, we develop a sieve maximum likelihood estimator for a general semiparametric class of ODE models. In comparison to existing estimation methods, the proposed procedure has advantages in terms of computational scalability and numerical stability. Moreover, to address unique theoretical challenges induced by the ODE notion, we establish a new general sieve M-theorem for bundled parameters and show that the proposed sieve estimator is consistent and asymptotically normal, and achieves the semiparametric efficiency bound. The finite sample performance of the proposed estimator is examined in simulation studies and a real-world data example. for this article are available online. [ABSTRACT FROM AUTHOR]

The stochastic block model is one of the most studied network models for community detection, and fitting its likelihood function on large-scale networks is known to be challenging. One prominent work that overcomes this computational challenge is the fast pseudo-likelihood approach proposed by Amini et al. for fitting stochastic block models to large sparse networks. However, this approach does not have convergence guarantee, and may not be well suited for small and medium scale networks. In this article, we propose a novel likelihood based approach that decouples row and column labels in the likelihood function, enabling a fast alternating maximization. This new method is computationally efficient, performs well for both small- and large-scale networks, and has provable convergence guarantee. We show that our method provides strongly consistent estimates of communities in a stochastic block model. We further consider extensions of our proposed method to handle networks with degree heterogeneity and bipartite properties. for this article are available online. [ABSTRACT FROM AUTHOR]

Abstract–The stochastic block model is widely used for detecting community structures in network data. How to test the goodness of fit of the model is one of the fundamental problems and has gained growing interests in recent years. In this article, we propose a novel goodness-of-fit test based on the maximum entry of the centered and rescaled adjacency matrix for the stochastic block model. One noticeable advantage of the proposed test is that the number of communities can be allowed to grow linearly with the number of nodes ignoring a logarithmic factor. We prove that the null distribution of the test statistic converges in distribution to a Gumbel distribution, and we show that both the number of communities and the membership vector can be tested via the proposed method. Furthermore, we show that the proposed test has asymptotic power guarantee against a class of alternatives. We also demonstrate that the proposed method can be extended to the degree-corrected stochastic block model. Both simulation studies and real-world data examples indicate that the proposed method works well. for this article are available online. [ABSTRACT FROM AUTHOR]