Abstract Nonresponse in probability sampling presents a long‐standing challenge in survey sampling, often necessitating simultaneous adjustments to address sampling and selection biases. We develop a statistical framework that explicitly models sampling weights as random variables and establish the semiparametric efficiency bound for the parameter of interest under nonresponse. This study investigates strategies for eliminating bias and effectively utilizing available information, extending beyond nonresponse issues to data integration with external summary statistics. The proposed estimators are characterized by their efficiency and double robustness. However, realizing full efficiency hinges on the accurate specification of underlying models. To enhance robustness against potential model misspecification, we expand double robustness to multiple robustness through a novel two‐step empirical likelihood approach. A numerical study evaluates the finite‐sample performance of our methods. Additionally, we apply these methods to a dataset from the National Health and Nutrition Examination Survey, effectively integrating summary statistics from the National Health Interview Survey.

Abstract In high dimensions, the relationship between covariates and a response variable becomes increasingly intricate, with different covariate components often displaying varying degrees of variability. This complex interplay of dependence, heterogeneity, and high dimensionality presents a significant challenge when investigating the effects of covariates on the response variable. To address this, we propose a novel marginal testing procedure based on kernel‐based conditional mean dependence, which can be implemented without requiring model assumptions. Theoretically, we establish the limiting normal distributions of the test statistic under both null hypotheses and local alternatives by asymptotically approximating a class of quadratic forms. We also examine the asymptotic relative efficiency of the proposed test against several state‐of‐the‐art alternatives. Our theoretical evaluations are conducted from two perspectives: a detailed analysis within a linear model framework and a comparison within a fully non‐parametric setting. The effectiveness and applicability of the proposed method are demonstrated through both simulation studies and real data analysis.

ABSTRACT Latent Gaussian models (LGMs) are a subset of Bayesian Hierarchical models where Gaussian priors, conditional on variance parameters, are assigned to all effects in the model. LGMs are employed in many fields for their flexibility and computational efficiency. However, practitioners find prior elicitation on the variance parameters challenging because of a lack of intuitive interpretation for them. Recently, several papers have tackled this issue by representing the model in terms of variance partitioning (VP) and assigning priors to parameters reflecting the relative contribution of each effect to the total variance. So far, the class of priors based on VP has been mainly applied to random effects and fixed effects separately. This work presents a novel standardization procedure that expands the applicability of VP priors to a broader class of LGMs, including both fixed and random effects. The practical advantages of standardization are demonstrated with simulated data and a real dataset on survival analysis.

Abstract Causal dependence modelling of multivariate extremes is intended to improve our understanding of the relationships among variables associated with rare events. Regular variation provides a standard framework in the study of extremes. This paper concerns the extremal causal dependence of the linear structural equation model with regularly varying noise variables. We focus on extreme observations generated from such a model and propose a causal discovery method based on the scaling parameters of its extremal angular measure. We implement the method as an algorithm, establish its consistency and evaluate it by simulation and by application to river discharge datasets. We propose a selection procedure for its hyperparameters based on a notion of stability. Comparison with the only alternative extremal method for such model reveals its competitive performance.

Abstract Model misspecification is ubiquitous in data analysis because the data‐generating process is often complex and mathematically intractable. Therefore, assessing estimation uncertainty and conducting statistical inference under a possibly misspecified working model is unavoidable. In such a case, classical methods such as bootstrap and asymptotic theory‐based inference frequently fail since they rely heavily on the model assumptions. In this article, we provide a new bootstrap procedure, termed local residual bootstrap, to assess estimation uncertainty under model misspecification for generalized linear models. By resampling the residuals from the neighboring observations, we can approximate the sampling distribution of the statistic of interest accurately. Instead of relying on the score equations, the proposed method directly recreates the response variables so that we can easily conduct standard error estimation, confidence interval construction, hypothesis testing, and model evaluation and selection. It performs similarly to classical bootstrap when the model is correctly specified and provides a more accurate assessment of uncertainty under model misspecification, offering data analysts an easy way to guard against the impact of misspecified models. We establish desirable theoretical properties, such as the bootstrap validity, for the proposed method using the surrogate residuals. Numerical results and real data analysis further demonstrate the superiority of the proposed method.

Abstract Modern computational advances have enabled easy parallel implementations of Markov chain Monte Carlo (MCMC). However, almost all work in estimating the variance of Monte Carlo averages, including the efficient batch means (BM) estimator, focuses on a single‐chain MCMC run. We demonstrate that simply averaging covariance matrix estimators from multiple chains can yield critical underestimates in small Monte Carlo sample sizes, especially for slow‐mixing Markov chains. We extend the work of Argon & Andradóttir (2006) and propose a multivariate replicated batch means (RBM) estimator that utilizes information from parallel chains, thereby correcting for the underestimation. Under weak conditions on the mixing rate of the process, RBM is strongly consistent and exhibits similar large‐sample bias and variance to the BM estimator. We also exhibit superior theoretical properties of RBM by showing that the (negative) bias in the RBM estimator is less than that of the average BM estimator in the presence of positive correlation in MCMC. Consequently, in small runs, the RBM estimator can be dramatically superior, and this is demonstrated through a variety of examples.