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A key challenge for modern Bayesian statistics is how to perform scalable inference of posterior distributions. To address this challenge, variational Bayes (VB) methods have emerged as a popular alternative to the classical Markov chain Monte Carlo (MCMC) methods. VB methods tend to be faster while achieving comparable predictive performance. However, there are few theoretical results around VB. In this article, we establish frequentist consistency and asymptotic normality of VB methods. Specifically, we connect VB methods to point estimates based on variational approximations, called frequentist variational approximations, and we use the connection to prove a variational Bernstein-von Mises theorem. The theorem leverages the theoretical characterizations of frequentist variational approximations to understand asymptotic properties of VB. In summary, we prove that (1) the VB posterior converges to the Kullback-Leibler (KL) minimizer of a normal distribution, centered at the truth and (2) the corresponding variational expectation of the parameter is consistent and asymptotically normal. As applications of the theorem, we derive asymptotic properties of VB posteriors in Bayesian mixture models, Bayesian generalized linear mixed models, and Bayesian stochastic block models. We conduct a simulation study to illustrate these theoretical results. Supplementary materials for this article are available online.

Networks are often characterized by node heterogeneity for which nodes exhibit different degrees of interaction and link homophily for which nodes sharing common features tend to associate with each other. In this article, we rigorously study a directed network model that captures the former via node-specific parameterization and the latter by incorporating covariates. In particular, this model quantifies the extent of heterogeneity in terms of outgoingness and incomingness of each node by different parameters, thus allowing the number of heterogeneity parameters to be twice the number of nodes. We study the maximum likelihood estimation of the model and establish the uniform consistency and asymptotic normality of the resulting estimators. Numerical studies demonstrate our theoretical findings and two data analyses confirm the usefulness of our model. Supplementary materials for this article are available online.

In survival analysis it often happens that some subjects under study do not experience the event of interest; they are considered to be “cured.” The population is thus a mixture of two subpopulations, one of cured subjects and one of “susceptible” subjects. We propose a novel approach to estimate a mixture cure model when covariates are present and the lifetime is subject to random right censoring. We work with a parametric model for the cure proportion, while the conditional survival function of the uncured subjects is unspecified. The approach is based on an inversion which allows us to write the survival function as a function of the distribution of the observable variables. This leads to a very general class of models which allows a flexible and rich modeling of the conditional survival function. We show the identifiability of the proposed model as well as the consistency and the asymptotic normality of the model parameters. We also consider in more detail the case where kernel estimators are used for the nonparametric part of the model. The new estimators are compared with the estimators from a Cox mixture cure model via simulations. Finally, we apply the new model on a medical data set.

Let \\(\\mathcal{T}\\) denote a Galton--Watson tree with offspring distribution \\(\\xi\\) satisfying \\(\\mathbb{E}(\\xi) = 1\\), and let \\(\\mathcal{T}_n\\) be the Galton--Watson tree conditioned to have exactly \\(n\\) nodes. We show that, under a mild moment condition on \\(\\xi\\), the number of occurrences of a fixed rooted plane tree \\(\\mathbf{t}\\) as a general subtree in \\(\\mathcal{T}_n\\) is asymptotically normal as \\(n \\to \\infty\\), with both mean and variance linear in \\(n\\). In addition, we prove that this limiting distribution is nondegenerate except for some special cases where the variance remains bounded. These results confirm a conjecture of Janson in recent work on the same topic. Finally, we present examples showing that if the proposed moment condition on \\(\\xi\\) is violated, the conclusion may fail.

An established and growing literature on generalized fiducial inference and related fiducial ideas points to the adoption of fiducial inference as a mainstream perspective among modern statisticians. Like Bayesian posteriors, generalized fiducial distributions (GFDs) are known to satisfy Bernstein-von Mises (BvM)-type results under classical regularity conditions. Existing fiducial BvM results, however, rely on relatively restrictive smoothness assumptions and are limited in scope. In this paper, we establish a Bernstein-von Mises theorem for generalized fiducial inference under the general framework of local asymptotic normality, which accommodates non-i.i.d. data settings and reduces to the familiar differentiability in quadratic mean condition in the i.i.d. case. We apply our result to extend existing fiducial theory for free-knot spline models first developed in Sonderegger and Hannig (2014), and further illustrate its generality in models where classical regularity conditions fail or i.i.d. assumptions are not met.

In this paper, we test the homogeneity of multiple covariance matrices when the dimension may exceed the sample sizes. A test statistic is proposed which does not depend on the normality assumption. Furthermore, the asymptotic distribution of the test statistic is derived. Numerical simulations indicate that the proposed test has accurate significance level, and has a greater improvement in power than some existing tests.

Mendelian randomization (MR) is a method of exploiting genetic variation to unbiasedly estimate a causal effect in presence of unmeasured confounding. MR is being widely used in epidemiology and other related areas of population science. In this paper, we study statistical inference in the increasingly popular two-sample summary-data MR design. We show a linear model for the observed associations approximately holds in a wide variety of settings when all the genetic variants satisfy the exclusion restriction assumption, or in genetic terms, when there is no pleiotropy. In this scenario, we derive a maximum profile likelihood estimator with provable consistency and asymptotic normality. However, through analyzing real datasets, we find strong evidence of both systematic and idiosyncratic pleiotropy in MR, echoing the omnigenic model of complex traits that is recently proposed in genetics. We model the systematic pleiotropy by a random effects model, where no genetic variant satisfies the exclusion restriction condition exactly. In this case, we propose a consistent and asymptotically normal estimator by adjusting the profile score.We then tackle the idiosyncratic pleiotropy by robustifying the adjusted profile score. We demonstrate the robustness and efficiency of the proposed methods using several simulated and real datasets.

Government policies during the COVID-19 pandemic have drastically altered patterns of energy demand around the world. Many international borders were closed and populations were confined to their homes, which reduced transport and changed consumption patterns. Here we compile government policies and activity data to estimate the decrease in CO2 emissions during forced confinements. Daily global CO2 emissions decreased by –17% (–11 to –25% for ±1σ) by early April 2020 compared with the mean 2019 levels, just under half from changes in surface transport. At their peak, emissions in individual countries decreased by –26% on average. The impact on 2020 annual emissions depends on the duration of the confinement, with a low estimate of –4% (–2 to –7%) if prepandemic conditions return by mid-June, and a high estimate of –7% (–3 to –13%) if some restrictions remain worldwide until the end of 2020. Government actions and economic incentives postcrisis will likely influence the global CO2 emissions path for decades.COVID-19 pandemic lockdowns have altered global energy demands. Using government confinement policies and activity data, daily CO2 emissions have decreased by ~17% to early April 2020 against 2019 levels; annual emissions could be down by 7% (4%) if normality returns by year end (mid-June).

We propose the Sequential Synthetic Difference-in-Differences (Sequential SDiD) estimator for event studies with staggered treatment adoption, particularly when the parallel trends assumption fails. The method uses an iterative imputation procedure on aggregated data, where estimates for early-adopting cohorts are used to construct counterfactuals for later ones. We prove the estimator is asymptotically equivalent to an infeasible oracle OLS estimator within a linear model with interactive fixed effects. This key theoretical result provides a foundation for standard inference by establishing asymptotic normality and clarifying the estimator's efficiency. By offering a robust and transparent method with formal statistical guarantees, Sequential SDiD is a powerful alternative to conventional difference-in-differences strategies.

The Bradley–Terry model is a fundamental model in the analysis of network data involving paired comparison. Assuming every pair of subjects in the network have an equal number of comparisons, Simons and Yao (Ann. Statist. 27 (1999) 1041–1060) established an asymptotic theory for statistical estimation in the Bradley–Terry model. In practice, when the size of the network becomes large, the paired comparisons are generally sparse. The sparsity can be characterized by the probability pn that a pair of subjects have at least one comparison, which tends to zero as the size of the network n goes to infinity. In this paper, the asymptotic properties of the maximum likelihood estimate of the Bradley–Terry model are shown under minimal conditions of the sparsity. Specifically, the uniform consistency is proved when pn is as small as the order of (log n)³/n, which is near the theoretical lower bound log n/n by the theory of the Erdős–Rényi graph. Asymptotic normality and inference are also provided. Evidence in support of the theory is presented in simulation results, along with an application to the analysis of the ATP data.