Abstract Gibbs posteriors are proportional to a prior distribution multiplied by an exponentiated loss function, with a key tuning parameter that weights the information in the loss relative to the prior and provides control of posterior uncertainty. Gibbs posteriors provide a principled framework for likelihood-free Bayesian inference; however, in many situations, the inclusion of a single tuning parameter inevitably leads to poor uncertainty quantification. In particular, regardless of the value of the parameter, credible regions are far from attaining nominal frequentist coverage, even in large samples. We propose a sequential extension to Gibbs posteriors to address this problem. We prove that the proposed sequential posterior exhibits concentration and satisfies a Bernstein–von Mises theorem, which holds under easily verifiable conditions in Euclidean space and on manifolds. As a by-product, we obtain the first Bernstein–von Mises theorem for traditional likelihood-based Bayesian posteriors on manifolds. All methods are illustrated with an application to principal component analysis.

Abstract Palm distributions play a central role in the study of point processes and their associated summary statistics. In this work, we characterize the Palm distributions of the superposition of independent point processes, establishing a simple mixture representation depending on the point processes’ Palm distributions and moment measures. We explore two statistical applications enabled by our main result. First, we consider minimum contrast estimation for corrupted point processes. Second, we investigate the class of shot noise Cox processes and derive explicit expressions for their higher-order Palm distributions. In the finite case, we further obtain a tractable expression for the Janossy density, which plays the role of a likelihood function and thus can be used for new likelihood-based inference strategies. Extensions to the superposition of multiple point processes and to higher-order Palm distributions are also presented.

Abstract Researchers often turn to block randomization to increase the precision of their inference or due to practical considerations, such as in multisite trials. However, if the number of treatments under consideration is large it might not be feasible or practical to assign all treatments within each block. We develop novel inference results under the finite-population design-based framework for natural alternatives to the complete block design that do not require reducing the number of treatment arms, the incomplete block design and the balanced incomplete block design. This includes deriving the properties of two design-based estimators, developing a finite-population central limit theorem, and proposing conservative variance estimators.Comparisons of the design-based estimators are made to linear model-based estimators. Simulations and a data illustration further demonstrate performance of incomplete block design estimators. This work highlights incomplete block designs as practical and currently underutilized designs.

Abstract The average treatment effect (ATE) is commonly used to quantify the main effect of a binary treatment on an outcome. Extensions to continuous treatments are usually based on the dose response curve or shift interventions, but both require strong overlap conditions and the resulting curves may be difficult to summarise. We focus instead on average derivative effects (ADEs) that are scalar estimands related to infinitesimal shift interventions requiring only local overlap assumptions. ADEs, however, are rarely used in practice because their estimation usually requires estimating conditional density functions. By characterising the Riesz representers of weighted ADEs,weproposeanewclassofestimandsthat provides a unified view of weighted ADEs/ATEs when the treatment is continuous/binary. We derive the estimand in our class that minimises the nonparametric efficiency bound, thereby extending optimal weighting results from the binary treatment literature to the continuous setting. We develop efficient estimators for two weighted ADEs that avoid density estimation and are amenable to modern machine learning methods, which we evaluate in simulations and an applied analysis of Warfarin dosage effects.

Summary Factorial experiments are ubiquitous in the social and biomedical sciences, but when units fail to comply with each assigned factor, identification and estimation of the average treatment effects become impossible without strong assumptions. Leveraging an instrumental variables approach, previous studies have shown how to define and estimate the causal effect of treatment uptake among respondents who comply with treatment. A major caveat is that these results rely on strong assumptions on the effect of randomization on treatment uptake. This work shows how to bound these complier average treatment effects for bounded outcomes under milder assumptions on noncompliance.

Abstract The estimation of regression parameters in spatially referenced data plays a crucial role across various scientific domains. A common approach involves employing an additive regression model to capture the relationship between observations and covariates, accounting for spatial variability not explained by the covariates through a Gaussian random field. We study the effect of misspecified covariates, in particular when the misspecification changes the smoothness. We analyse the theoretical properties of the generalized least-squares estimator under infill asymptotics, and show that the estimator can have counter-intuitive properties. In particular, the estimated regression coefficients can converge to zero as the number of observations increases if the covariates are too rough, despite high correlations between observations and covariates. This has important implications for practical applications as the importance of rough covariates can be severely underestimated, leading to incorrect scientific conclusions. We also show that the estimates can diverge to infinity under certain conditions, which can also lead to incorrect conclusions in practical applications. Through an application to temperature and precipitation data, we show that both behaviours can be observed for real data. Finally, we propose adding a smoothing step in the regression and show both theoretically and practically that this can solve the problem.

Abstract We propose a method for comparing survival data based on higher criticism of $ p $-values obtained from many exact hypergeometric tests. The method accommodates noninformative right-censorship and is sensitive to hazard differences in unknown and relatively rare time intervals. It attains much better power against such differences than the log-rank test and its variants. We demonstrate the usefulness of our method in detecting rare and weak non-proportional hazard differences compared to existing tests using simulations and gene expression data. Additionally, we analyse the asymptotic power of our method and other tests under a theoretical framework describing two groups experiencing failure rates that are usually identical over time, except in a few unknown instances where one group’s failure rate is higher. Our test’s power experiences a phase transition across the plane of rarity and intensity parameters that mirrors the phase transition of higher criticism in two-sample rare and weak normal and Poisson means settings. The region of the plane in which our method has asymptotically full power is larger than the corresponding region for the log-rank test.

Summary Asymptotically unbiased priors, introduced by Hartigan (1965), are designed to achieve second-order unbiasedness of Bayes estimators. This paper extends Hartigan’s framework to non-independent-and-identically-distributed models by deriving a system of partial differential equations that characterizes asymptotically unbiased priors. Furthermore, we establish a necessary and sufficient condition for the existence of such priors and propose a simple procedure for constructing them. The proposed method is applied to the linear regression model and the nested error regression model (also known as the random effects model). Simulation studies evaluate the frequentist properties of the Bayes estimator under the asymptotically unbiased prior for the nested error regression model, highlighting its effectiveness in small-sample settings.

Abstract Estimating the mean square error of a small area predictor under an informative sampling design is a challenging problem. Existing approaches rely on approximations that have not been justified theoretically. We provide rigorous support for a mean square error estimator that is applicable to an informative sample design. The procedure can be used in combination with predictors of general parameters that may be nonlinear functions of the model response variable. We also construct calibrated prediction intervals that rely less on normality than standard prediction intervals. We validate the proposed measures of uncertainty through simulation. We apply the methods to predict several functions of sheet and rill erosion for Iowa counties using data from a complex agricultural survey.

Summary This paper introduces novel weighted conformal p-values and methods for model-free selective inference. The problem is as follows: given test units with covariates X and missing responses Y, how do we select units for which the responses Y are larger than user-specified values while controlling the proportion of false positives? Can we achieve this without modelling assumptions on the data and without restriction on the model for predicting the responses? Last, methods should be applicable when there is a covariate shift between training and test data, which commonly occurs in practice. We answer these questions by first leveraging any prediction model to produce a class of well-calibrated weighted conformal p-values, which control the type-I error in detecting a large response. These p-values cannot be passed onto classical multiple testing procedures since they may not obey a well-known positive dependence property. Hence, we introduce weighted conformal selection, a new procedure that controls the false discovery rate in finite samples. Besides prediction-assisted candidate selection, weighted conformal selection (i) allows us to infer multiple individual treatment effects, and (ii) extends to outlier detection with inlier distribution shifts. We demonstrate performance via simulations and applications to causal inference, drug discovery and outlier detection datasets.