ShrinkageTrees is an R package for Bayesian tree ensembles in survival analysis and causal inference. The package implements Bayesian additive regression tree models for right- and interval-censored survival outcomes within an accelerated failure time (AFT) framework, with optional decomposition into prognostic and treatment-effect components for causal inference. Two complementary forms of regularisation are available: regularisation of the tree structure, via depth-penalising priors and Dirichlet splitting priors, and regularisation of the step heights, via global-local shrinkage priors. ShrinkageTrees provides the first implementation of the Horseshoe Forest, which places a horseshoe prior on the step heights. These regularisation strategies extend Bayesian tree ensembles to high-dimensional settings. An efficient Rcpp backend, multi-chain MCMC, and S3 methods support the full workflow: fitting, prediction, causal effect estimation, and convergence diagnostics.

Preference learning refers to the learning of latent patterns from ranking and preference data of different kinds. Typical aims of preference learning are to infer a shared consensus ranking, to learn individual-level preferences, and to perform unsupervised clustering. The Mallows model is among the few approaches that can achieve all these objectives jointly. Previous work has developed computationally tractable methods for Bayesian inference based on a MCMC Metropolis-Hastings scheme, where clustering is performed via a finite mixture of Mallows models. Inference on the number of clusters is then conducted a posteriori. Here we propose a Bayesian nonparametric Mallows model, based on a Dirichlet process mixture model. This allows joint inference on the number of non-empty clusters and on the clustering allocation, as well as posterior inference on cluster-specific parameters. The implementation of the proposed sampling algorithm is integrated into the existing R package BayesMallows, which also supports data in the form of incomplete rankings and pairwise comparisons. Simulated data show good performance of the nonparametric model compared to a finite mixture model in terms of recovery of the correct number of clusters, while empirical data on movie ratings show the model's effectiveness in providing personalized movie recommendations on discarded ratings.

Conventional nonparametric regression methods for two-dimensional non-rectangular domains often overlook domain geometry and allow smoothing across boundaries. In spatial and geostatistical applications, this assumption is frequently invalid because domain boundaries typically constrain interactions among observations. Accommodating spatially varying smoothness is also substantially more challenging than in the univariate setting, and most existing methods do not adequately capture this local structure of the target function. To address these challenges, we propose Bayesian triangulation splines, which constructs locally adaptive splines over a polygonal domain. The method employs constrained Delaunay triangulations to respect boundary geometry and adapt to heterogeneous smoothness. A carefully designed prior further improves empirical performance. Under a global Sobolev smoothness assumption, we show that the proposed method achieves the optimal posterior contraction rate and adapts to unknown smoothness. We also show that the method exhibits ideal spatial adaptation in the sense that it achieves the oracle rate for inhomogeneous or locally varying structural features. Crucially, this oracle guarantee is not specific to constrained Delaunay triangulations, but holds over any triangulation satisfying weak shape-regularity conditions. Simulation studies confirm that the proposed method outperforms existing approaches by achieving higher estimation accuracy while maintaining low model complexity.

Quantifying the likelihood of extreme events underpins risk assessment, yet classical Extreme Value Theory relies on asymptotic assumptions that fail in the data-sparse, non-stationary regimes practitioners increasingly face. We introduce the Data-Driven Extreme Value Distribution (DDEVD), a non-parametric estimator that aggregates all observations metastatistically and reconstructs the base distribution with a kernel, removing parametric tail assumptions. We derive its optimal bandwidth and prove a stability law $m < C\,n^{1+\gamma/2}$ relating reliable extrapolation to the extreme value index $\gamma$. In sub-hourly Alpine precipitation, DDEVD recovers stable 100-year return levels from single decades (calibration ratio $0.96$), departing from the full-record reference by over $50\,\%$ in fewer than one window in fifty -- versus one in five for a GEV fit. In metallurgical micrographs, it matches a generalised extreme-value fit on the safety-relevant grain-size tail, where the standard log-normal over-predicts by $58\,\%$ at $1\,\mathrm{cm}^{2}$.

Air pollution regulatory limits are typically defined in terms of exceedances of concentration thresholds which are naturally related to conditional quantiles of the pollutant distribution and are therefore of direct relevance for assessing severe pollution events. At the same time, it is important to determine not only whether a covariate affects air pollution but also whether this effect is linear, nonlinear, or both. We address these issues by developing a Bayesian effect selection approach for additive quantile regression. While commonly used mixed model representations (MMRs) of penalized splines allow for flexible nonlinear effects, they do not provide a meaningful separation of linear and nonlinear effect components. We therefore employ a Demmler-Reinsch basis expansion, which yields an orthogonal decomposition of each additive effect into linear and nonlinear parts and show theoretically that both effect components can be estimated consistently. To facilitate data-driven model building, we propose Bayesian effect selection with separate spike and slab priors on the scalar importance parameters associated with the linear and nonlinear components and implement an efficient Gibbs sampler. Through simulation studies, we demonstrate robustness to the misspecification induced by the employed asymmetric Laplace working likelihood and show superior performance relative to the MMR. In a detailed analysis of air pollution data in Madrid, Spain we highlight the added value of flexibly modeling extreme nitrogen dioxide (NO$_2$) concentrations and reveal that threshold-relevant pollution levels are driven differently by climatological variables and traffic-related spatial structure. These findings underline the need for advanced statistical models that support short-term decision-making and help local authorities mitigate, or potentially prevent, exceedances of NO$_2$ concentration limits.

Ripley's K-function is a widely used spatial summary statistic for assessing clustering in point patterns. However, existing K-based methods can be computationally prohibitive for large-scale data, particularly in high-throughput spatial proteomics, because they rely on spatial information from all points in the image. To address this challenge, we propose a computationally efficient block-based testing framework that extracts disjoint local blocks from an image and aggregates clustering evidence across them. The proposed adaptive spatial blocking algorithm constructs blocks satisfying point-count and shape constraints, enabling scalable spatial clustering inference and fast p-value computation through an asymptotic normal approximation. Numerical studies demonstrate that the proposed method provides a favorable balance between statistical power and computational efficiency. In an application to healthy human intestine spatial proteomics data, our method detects strong spatial aggregation of plasma cells and colocalization between plasma cells and macrophages, while scaling favorably to large images.

We propose a novel, intuitive measure of statistical performance: precision-weighted bias. Precision-weighted bias is defined as the unconditional bias of an estimator weighted by the degree of information (precision) it contains. Current guidelines, such as GRADE and CONSORT, often view the potential for increased bias in adaptive designs as a deterrent for the inclusion of such designs in systematic reviews. However, we demonstrate that the bias in a common-effect meta-analysis is approximately equal to the precision-weighted average of the precision-weighted biases of its constituent studies, rather than of their unweighted unconditional biases. Through simulation studies, we show that while adaptive designs may exhibit unweighted bias, they frequently have zero precision-weighted bias. Consequently, including these designs often results in a negligible change to the overall meta-analysis bias. These results suggest that precision-weighted bias is a superior indicator for determining whether to include an adaptive design in a meta-analysis. We recommend that precision-weighted bias be used as a standard complement to unweighted unconditional and conditional bias in simulation studies to support more inclusive and accurate evidence synthesis.

The composite likelihood method reduces the computational cost of parameter estimation in time series by considering several subsets of observations instead of all observations at once. The asymptotic properties of this method are related to the Godambe information, an extension of the Fisher information that accounts for the dependence between subsets of observations. We aim to apply this method to linear Gaussian models, in particular fractional Brownian motion and fractional Gaussian noise. We derive theoretical expressions for their Fisher information and their Godambe information and deduce a subset selection design that sequentially maximizes the Godambe information. The size of the subsets then allows us to control the trade-off between estimation accuracy and computational cost. Through simulations, we compare this method with the method of moments and maximum likelihood estimation, and we apply it to real data, namely volatility series of a stock index and a wind speed time series.

The article deals with the problem of testing conditional axial symmetry within a~multivariate linear heteroscedastic regression framework. A new test based on integrated rank scores is introduced and its asymptotic distribution is derived. The proposed method extends a similar procedure developed for multivariate data to the regression setting. The test may also be employed to assess specific hypotheses concerning distributional properties of the error term. Its performance and application is illustrated in a small simulation study and with real economic data. The article also contains a few theoretical results regarding axial symmetry that may be of independent interest.

This work presents a new method for principal component analysis (PCA) of a mixed-type data consisting of continuous, binary, integer-valued and positive continuous variables. The data are assumed to come from a probability model, where the parameters of the exponential family distributions are determined by a set of shared Gaussian latent variables. The proposed method, MTPCA, is based on estimating the covariance matrix of these latent mixtures through the method of moments. A way to sparsify the component loadings is presented and aligns with the classical theory of sparse PCA. We propose a strategy for estimating the principal component scores and discuss the choice of the latent dimension. The method's performance is studied with a simulated mixed-type data and we illustrate the model on the Zoo data set consisting of binary animal characteristics.

Circulating-tumour DNA (ctDNA) carries evidence of drug resistance months before imaging shows it, but the earliest evidence lives below the assay's limit of detection (LoD): a nascent subclone is detected only intermittently, producing a flickering sequence of faint detects and non-detects. Commercial liquid biopsies treat each draw as an independent snapshot and a non-detect as nothing. We argue a non-detect is a left-censored observation, and the pattern of non-detects and faint detects over time carries actionable evidence of growth before any single value is trustworthy. We introduce Span, a censored-Poisson Bayesian latent-growth change-point detector that models the binary detection process, accumulates a sequential generalised-likelihood-ratio statistic for an upward change-point in the per-variant detection rate, and raises a competing-risks alarm with calibrated false-alarm control. Span has no learned weights, so there is nothing to overfit. On a synthetic cohort of HR+/HER2- metastatic breast cancer on first-line CDK4/6-inhibitor plus endocrine therapy, at a matched 10% false-alarm rate, Span roughly doubles the fraction of impending progressions caught three months ahead (indolent regime: 25% vs 11% for the snapshot), with a falsifiable dose-response: large for indolent emergence, vanishing for fast emergence. A value-trajectory baseline performs identically to the snapshot, isolating the gain to the censored detection model. The survival backbone matches a Cox baseline on real breast-cancer data (GBSG-2, n=686; C-index 0.67 vs 0.68), and on a real longitudinal cohort with clean biomarkers (PBC2, n=312) the same pipeline correctly declines to win, a falsifiable boundary test confirming the mechanism is regime-specific. All ctDNA trajectories are synthetic.

We introduce the Hierarchical Excitatory Process (HEP), a flexible point process model for event-time data observed under repeated external stimuli. The proposed framework models the conditional intensity of a point process as a superposition of excitation effects induced by external stimuli, characterised by kernels with parameters dynamically evolving over time. This hierarchical construction enables modulation of excitation strength across repeated stimuli, providing an interpretable structure. We establish likelihood-based inference for the proposed model and embed HEP within a model-based clustering framework to identify latent groups sharing similar response dynamics. Simulation studies demonstrate the model's ability to recover evolving latent patterns, and an application to spike train recordings from the sea slug Aplysia pedal ganglion illustrates how HEPs are able to characterise stimulus-driven excitability of neurons across repeated stimulation under different experimental conditions.

Under the canonical setting of observational studies for causal inference, we show that the average treatment effect under the overlap weight, the weight that is proportional to the conditional variance of the treatment given the covariates, is bounded between the average treatment effects on the treated and control, under a monotonic relationship between the propensity score and the conditional average treatment effect. We further extend the result to weighted local average treatment effects, under the canonical setting with a binary instrumental variable and a binary treatment. We also extend the results to other weights. Based on the theory, we recommend the ``CP-plot'' of the estimated conditional average treatment effect against the estimated propensity score.

The beta distribution is the default choice of likelihood in many regression problems with a $[0,1]$-bounded support response despite its sensitivity to outliers, inability to accommodate exact zero observations, and a lack of closed-form conjugate priors. We address these shortcomings by introducing the triply-randomized negative binomial beta distribution, denoted $\mathrm{TNBbeta}(p,\,q,\,\varepsilon)$, parameterized by a median $p$, concentration parameter $q$, and boundary parameter $\varepsilon$ which permits positive density at $0$ and $1$. The TNBbeta arises by randomizing the parameters of a standard beta distribution with three dependent negative binomial random variables, each of whose complete conditional distribution we show is itself negative binomial. Moreover, connecting $p$ and $q$ to Gaussian latent variables with logit link functions yields closed-form updates via Pólya-gamma augmentation. Together, these properties yield simple auxiliary-variable Gibbs samplers for regression models of bounded-support data, which often outperform standard beta regression approaches in terms of effective sample size per second and held-out prediction, especially in the presence of outliers. In a case study of forest canopy cover, we demonstrate that this framework can easily incorporate spatial structure and exact zero observations. Overall, this work substantially expands the class of Bayesian models for $[0,1]$-bounded support data that can be fit efficiently.

We propose a spectral-based, unsupervised representation learning framework to derive low-dimensional embeddings for clinical concepts and patients in rare disease cohorts from electronic health records, where data are high-dimensional but sample sizes are limited. To overcome this challenge, we incorporate a knowledge matrix extracted from a broader population that shares a partially overlapping subspace with the rare-disease cohort. Our method departs from existing approaches by relaxing restrictive one-to-one signal-alignment assumptions between the latent data matrix and knowledge matrix, allowing more flexible and realistic forms of structured sharing. We introduce a novel two-step spectral embedding procedure: first, we identify and remove irrelevant components from the knowledge matrix; then, we apply a projection-based method to separately recover shared and heterogeneous components. Simulations and an analysis of a real-world multiple sclerosis cohort show that the proposed method outperforms competing approaches, particularly in challenging scenarios where shared signals are weak and only partially aligned, as is common in rare-disease data.

This paper is concerned with estimating the local false discovery rate (lfdr) in a two-groups model where the only assumption regarding the null distribution is symmetry about zero. Our motivation comes from the contemporary framework for multiple hypothesis testing, particularly relevant in variable selection problems, which transforms any user-specified scores into statistics whose null distributions are symmetric about zero, whereas enrichment to the right of zero is generally expected for the non-nulls. While modern methods such as the knockoff filter (Barber and Candes; 2015) are able to exploit the null property for controlling the false discovery rate (FDR), an arguably more appropriate goal is to target control of the local false discovery rate for the rejected hypotheses, as proposed in Soloff et al. (2024) where the standard two-groups model (known $f_0$ and independence) is analyzed. Here, we take a step in this direction and propose to estimate the lfdr by targeting the surrogate density ratio $f(-w)/f(w)$, for $w>0$, where $f$ is the marginal density in the aforementioned ``stripped-down'' two-groups model. We study several estimators and propose a logistic regression based method with natural cubic spline basis. We also show that any consistent estimator of this surrogate yields asymptotic lfdr control of the multiple testing procedure that thresholds the estimate at the nominal level.

We describe Chronos LTV, a system to measure the long-term impact of delays and other service defects on key business metrics. We use Markov decision processes to model customer interactions over time, and formalize our target estimand as the marginal policy effect with respect to moving the average delay rate. Given this setup, we show that we can identify long-term effects under a sequential unconfoundedness assumption where delays are as good as random given observed order characteristics; and can estimate these effects using a simple covariate-balancing algorithm.

Interval-censored data frequently arise in clinical research where event times are only known to fall within specific assessment windows. Although the Cox proportional hazards model is a standard approach for such data, existing Wald-type tests often suffer from instability or poor performance in small samples. In this paper, we propose a robust spline-sieve-based likelihood ratio test for interval-censored data. We develop a computationally efficient estimation framework that ensures numerical stability. Furthermore, we rigorously establish the asymptotic distribution of the proposed likelihood ratio statistic, providing a solid theoretical foundation for statistical inference. Extensive simulation studies demonstrate that our approach achieves superior error control and higher power compared with traditional approaches. The practical utility of the method is further illustrated through the analysis of a real-world clinical dataset.

We prove that optimally weighted second-order least squares (SLS) and the degree-two generalized polynomial maximization method (PMM) are the same population estimating equation for linear regression with conditionally homoskedastic non-Gaussian errors: they choose the same optimal linear combination of the first two centered residual moments, solve one population normal system, share one influence function, and attain the common asymptotic variance $c_2g_2/N$ -- the ordinary-least-squares slope-variance factor $c_2$ scaled by the PMM variance-reduction coefficient $g_2=1-\gamma_3^2/(2+\gamma_4)$ (with $\gamma_3,\gamma_4$ the error skewness and excess kurtosis). Feasible plug-in implementations are therefore first-order equivalent, with only higher-order finite-sample differences. The identity is sharp: under heteroskedasticity the unconditional PMM body and the conditional SLS weighting separate, costing efficiency for symmetric errors and consistency for asymmetric errors. Beyond degree two, PMM holds an efficiency reserve that SLS cannot reach within its second-moment span. For symmetric platykurtic errors SLS collapses to ordinary least squares for the slope, while degree-three PMM exploits kurtosis information outside the SLS moment span through a closed-form coefficient $g_3$; for canonical asymmetric laws this reserve is $30$--$50\%$ within the degree-three polynomial moment class. The Lean 4 development machine-checks the degree-specific algebraic core -- the closed forms for $g_2$ and $g_3$, the $g_2\le1$ result, the design cancellations, and the symmetric collapse -- while the general monotonicity $g_{S+1}\le g_S\le1$ is proved analytically by nesting. A Monte Carlo study illustrates the equivalence, the reserve, and the heteroskedastic boundary at finite samples.

We propose a novel method that simultaneously determines the sample size for testing two survival probabilities at a pre-specified ltime while guaranteeing type I error control in both fixed and group-sequential trial designs. Simulations across varying hypothesized differences, failure distributions, censoring proportions, and nominal powers demonstrate consistent performance, while interim analyses highlight reduced type I error and increased power at each look, regardless of the underlying failure time distribution or spending function. Importantly, our method is especially useful for evaluating survival outcomes at a fixed time in randomized trials where one treatment arm includes neoadjuvant therapy prior to surgery while the other involves surgery alone. Furthermore, it is advantageous when the proportional hazards assumption is not satisfied, as often occurs in immunotherapy trials with delayed or time-varying treatment effects or crossing survival curves. The method is also applicable to randomized phase II trials, where smaller sample sizes and the use of intermediate or surrogate time-to-event endpoints demand efficient data use and robust error control. We illustrate the approach with motivating examples in renal and prostate cancer. An accompanying R Shiny application enables investigators to compute sample sizes interactively, facilitating practical trial planning in diverse settings.

In survival studies, treatments can benefit patients through different mechanisms: a treatment may increase the probability of being cured or delay failure among patients who are not cured. Quantifying which mechanism is dominant, and whether it varies across subpopulations, is clinically important, yet there is limited work in the causal machine learning literature addressing this problem. Standard causal survival learners target finite-horizon survival or restricted mean survival time, while many cure models capture cure structures without estimating causal effects. In this work, we define meaningful causal effects in the presence of a cured subpopulation and introduce BartCure, a Bayesian causal machine learning approach for estimating them. The causal effects we recommend decompose the causal effect on restricted mean survival time into a stochastic cure and stochastic latency component, and we relate these new effects to both stochastic intervention effects and causal effects in principal strata. In simulations, BartCure is competitive for estimating average effects and is especially effective at conservatively detecting the direction of treatment-effect heterogeneity. We apply BartCure to estimate average and subgroup causal effects and to identify treatment effect heterogeneity in the CALGB 40101 breast cancer trial.

A key step in knowledge discovery is the evaluation of data mining results. In several applications, including pattern mining, graph analysis, and others, this step includes the evaluation of the statistical significance of the results, to avoid spurious discoveries due only to noise or random fluctuations in the data. While specialized procedures have been developed for some specific applications, resampling-based approaches are widely used, in particular for complex analyses where analytical results cannot be derived. However, current resampling-based approaches require the generation and analysis of thousands of resampled datasets, and are therefore impractical for large datasets or computationally intensive analyses. In this paper, we introduce FewRS, a simple and effective resampling-based approach to assess the statistical significance of data mining results with rigorous guarantees on the probability of false discoveries. Our approach can be used in every situation where resampling-based approaches are applied. FewRS builds on our derivation of a novel bound to the supremum deviation of test statistics representing the quality of data mining results. We prove that FewRS needs to generate and analyze an extremely small number of resampled datasets, leading to a highly scalable approach with wide applicability. We test our approach on common tasks such as pattern mining and network analysis. In all cases, our approach results in a reduction of up to two orders of magnitude in running time compared to the state of the art, while preserving high statistical power, enabling the statistical validation of data mining results on large-scale real-world datasets.

This paper presents closed BH, a uniform improvement of the False Discovery Rate controlling method of Benjamini and Hochberg (BH). Closed BH is valid under the same assumption of Positive Regression Dependency on a Subset (PRDS) as BH, but also under an alternative and weaker minimal sufficient condition. As a uniform improvement, closed BH never rejects fewer hypotheses than BH, but it may reject quite a few more. An increase in power is observed especially when the number of false null hypotheses is large. The novel method is constructed using the e-Closure principle, a recently derived general principle for multiple testing. The method is implemented in the eClosure package in R.

We consider the problem of estimating the true Sharpe ratio of an asset selected for having the highest observed in-sample Sharpe ratio among many assets. We discuss estimators based on the polyhedral lemma, James Stein shrinkage, debiasing the expected maximum Sharpe ratio, thresholding and empirical Bayes. We test these estimators in simulations, computing bias and root mean square error across different values of sample size, number of assets, and spread and shape of population Sharpe ratios. We also compute rank correlation of the estimators against the underlying quantity, simulating how these estimators might be used to compare or rank the output of different teams which perform this selection process. We find that the James Stein estimator provides the best performance across many different realistic values of the relevant parameters, followed by the GMLEB estimator of Jiang and Zhang. These results are fairly robust to correlation of asset returns, with some caveats.

Generalized partially linear single-index additive models (GPLSIAMs) have been increasingly applied across diverse areas due to their versatility in integrating functional flexibility with parametric dimension reduction while maintaining interpretability. However, the estimation presents severe computational challenges. This paper introduces a novel stable method that uses the model matrix for each single-index effect, defined by its single-index coefficients, and the penalized complete Fisher information matrix to dynamically update the boundaries of the single-index covariates within a unified iterative framework. The derived model matrices enable the fast computation of the estimated effective degrees of freedom and pointwise confidence bands for the single-index effects. The smoothing parameter updates are integrated into the iterative process via the generalized Fellner-Schall method, which recycles the derived matrix decompositions, thereby providing an efficient approximation to the global penalized optimization problem. Simulation studies with moderate sample sizes under non-Gaussian distributions confirm the empirical consistency of the estimation across multiple scenarios. Notably, the proposed approach remains stable where state-of-the-art competitive methods fail to recover true single-index coefficients and nonlinear functions, and is 80.13 times faster than the usual two-step method in the most computationally intensive scenario. The modeling advantage is illustrated through an application to Capital Bike Sharing data, where we deal with a single-index interaction effect for each year, with distinct single-index coefficients, a complex structure that makes competitive methods inapplicable. The proposed method is implemented in R, with functions available for reproducibility and transparency in comparisons.

We study causal inference in experiments and quasi-experiments, where the economic outcome is imperfectly measured by a remotely sensed variable. The remotely sensed variable is low-cost, scalable, and predictive of the economic outcome in observational data; examples include satellite imagery and mobile phone activity. We model the remotely sensed variable as post-outcome: variation in the economic outcome causes variation in the remotely sensed variable. For example, changes in environmental quality cause changes in satellite imagery, not vice versa. Under this assumption, we propose a formula to nonparametrically identify the causal parameter by combining experimental and observational data. We develop a method for n^{-1/2} inference that is robust to misspecification and that does not restrict the algorithms used to process remotely sensed variables.

Many real-world networks exhibit hierarchical, tree-like structure and heavy-tailed degree distributions, phenomena not readily captured by standard statistical models for network data. Extensions of the popular continuous latent space modeling framework have been proposed to accommodate such networks. Drawing on insights from statistical physics, continuous latent space models with underlying hyperbolic geometry have been proposed as a natural framework, probabilistically embedding nodes in a latent Riemannian manifold with constant negative curvature. Most statistical implementations, however, simplify the original physics-based model by omitting the ``temperature parameter," which controls the sharpness of the latent distance-to-probability mapping. We argue this omission is critical. We demonstrate that temperature is the fundamental parameter governing a network's tree-like topology, and that failing to infer it weakens model expressiveness. We formalize a Bayesian hyperbolic continuous latent space model with an unknown, learnable temperature parameter. We then develop two inferential procedures: a Hamiltonian Monte Carlo approach for rigorous posterior characterization and a scalable auto-encoding variational Bayes algorithm for large-scale networks. Through simulation and real data examples, we show that our model outperforms models with fixed temperature and misspecified Euclidean geometries in graph reconstruction tasks in most settings, confirming temperature is a crucial and inferable feature of complex networks.

Subnational monitoring of public health and development indicators often relies on household surveys where data are sparse at the desired spatial resolution. Small area estimation (SAE) methods address this challenge by borrowing strength across areas and incorporating auxiliary information. However, comparing these estimators remains difficult in the absence of ground truth. We propose a design-based cross-validation framework for evaluating small area estimators that accommodates complex survey designs. Our approach enables model-agnostic comparisons between area-level and unit-level SAE models. We derive a decomposition of the conditional mean squared error that yields a consistent cross-validation score, show that finite-sample comparisons carry an unidentifiable bias that can be bounded, and use this bound as a principled threshold for ranking models. We further show that leave-one-area-out cross-validation, a popular alternative, targets extrapolation rather than smoothing error and can reverse the correct ranking. We evaluate the framework through extensive design-based simulations. We apply the framework to compare subnational female literacy estimators in Zambia using the 2024 Demographic and Health Survey. The framework applies broadly across prevalence mapping and other SAE problems and is applicable to any small area estimator irrespective of the underlying model class.

The empirical Bayes $g$-modeling approach based on the nonparametric maximum likelihood estimator (NPMLE) has been central to large-scale estimation and inference in the normal means problem. However, theoretical guarantees for uncertainty quantification remain scarce. A key obstacle is that the NPMLE is necessarily discrete, which yields discrete posterior credible sets and a slow logarithmic deconvolution rate. We address both limitations by introducing a hierarchical Gaussian smoothing layer that restricts the mixing distribution to a Gaussian location mixture. Our smooth NPMLE inherits the favorable properties of the classical NPMLE: it is computable via convex optimization and achieves nearly parametric denoising performance. Moreover, it achieves a polynomial deconvolution rate that is asymptotically minimax over the corresponding class. Our procedure also leads to estimated smooth posteriors that converge to the true posteriors at a polynomial rate. Further, we characterize marginal coverage sets that are optimal in expected length, construct plug-in estimators of these sets, and establish theoretical guarantees for the estimated sets in terms of both coverage probability and expected length. We also extend the theory to settings with model misspecification and heteroscedastic Gaussian observations, and study identifiability of the proposed hierarchical model.

The Cauchy combination test (CCT) is widely used because it yields a closed-form combined $p$-value and is known to be asymptotically valid as the nominal level $\alpha\downarrow0$ under broad dependence structures. We study a different asymptotic question: whether the usual Cauchy cutoff remains accurate at an ordinary fixed level when the number $K$ of combined $p$-values grows under dependence. Under a canonical one-factor equicorrelated Gaussian copula model, we show that the raw CCT is generally not asymptotically exact at fixed $\alpha$. With fixed positive correlation, the statistic converges to a random latent-factor limit, so there is no universal fixed-level reference law. When the common correlation $\rho_K$ weakens with $K$, fixed-level behaviour is governed by the boundary-layer scale $s_K=\sqrt{\rho_K}(\log K)^{3/2}$, and the raw CCT is asymptotically exact if and only if $\rho_K(\log K)^3\to0$. Because the size distortion arises entirely from the reference law and not from the statistic, it can be corrected without modifying the test statistic itself. We propose the boundary-layer calibrated CCT (BL-CCT), which replaces the standard Cauchy reference by a one-parameter Gaussian-smoothed Cauchy family. Unlike recent variants that modify the test statistic, BL-CCT leaves the statistic unchanged and corrects only the reference law. BL-CCT is asymptotically exact under the weaker condition $\rho_K\log K\to0$ and provides a useful finite-$K$ approximation on bounded boundary layers. We also conduct several power analyses: although BL-CCT only raises the cutoff, it incurs no first-order power loss relative to the raw CCT on the exactness scale, under local dense, sparse, and dense Gaussian alternatives. Numerical experiments support the calibration theory.