ArXiv Paperboy (Stat.ME+Econ.EM)

When conducting inference for the average treatment effect on the treated with a Synthetic Control Estimator, the vector of control weights is a nuisance parameter which is often constrained, high-dimensional, and may be only partially identified even when the average treatment effect on the treated is point-identified. All three of these features of a nuisance parameter can lead to failure of asymptotic normality for the estimate of the parameter of interest when using standard methods. I provide a new method yielding asymptotic normality for an estimate of the parameter of interest, even when all three of these complications are present. This is accomplished by first estimating the nuisance parameter using a regularization penalty to achieve a form of identification, and then estimating the parameter of interest using moment conditions that have been orthogonalized with respect to the nuisance parameter. I present high-level sufficient conditions for the estimator and verify these conditions in an example involving Synthetic Controls.

arXiv📈🤖
Robust Inference when Nuisance Parameters may be Partially Identified with Applications to Synthetic Controls
By Fry

arXiv:2411.10648v1 Announce Type: new Abstract: Testing for mediation effect poses a challenge since the null hypothesis (i.e., the absence of mediation effects) is composite, making most existing mediation tests quite conservative and often underpowered. In this work, we propose a subsampling-based procedure to construct a test statistic whose asymptotic null distribution is pivotal and remains the same regardless of the three null cases encountered in mediation analysis. The method, when combined with the popular Sobel test, leads to an accurate size control under the null. We further introduce a Cauchy combination test to construct p-values from different subsample splits, which reduces variability in the testing results and increases detection power. Through numerical studies, our approach has demonstrated a more accurate size and higher detection power than the competing classical and contemporary methods.

arXiv📈🤖
Subsampling-based Tests in Mediation Analysis
By

arXiv:2401.14359v2 Announce Type: replace Abstract: This paper introduces several enhancements to the minimum covariance determinant method of outlier detection and robust estimation of means and covariances. We leverage the principal component transform to achieve dimension reduction and ultimately better analyses. Our best subset selection algorithm strategically combines statistical depth and concentration steps. To ascertain the appropriate subset size and number of principal components, we introduce a bootstrap procedure that estimates the instability of the best subset algorithm. The parameter combination exhibiting minimal instability proves ideal for the purposes of outlier detection and robust estimation. Rigorous benchmarking against prominent MCD variants showcases our approach's superior statistical performance and computational speed in high dimensions. Application to a fruit spectra data set and a cancer genomics data set illustrates our claims.

arXiv📈🤖
Minimum Covariance Determinant: Spectral Embedding and Subset Size Determination
By

Economic and financial time series can feature locally explosive behavior when a bubble is formed. The economic or financial bubble, especially its dynamics, is an intriguing topic that has been attracting longstanding attention. To illustrate the dynamics of the local explosion itself, the paper presents a novel, simple, yet useful time series model, called the stochastic nonlinear autoregressive model, which is always strictly stationary and geometrically ergodic and can create long swings or persistence observed in many macroeconomic variables. When a nonlinear autoregressive coefficient is outside of a certain range, the model has periodically explosive behaviors and can then be used to portray the bubble dynamics. Further, the quasi-maximum likelihood estimation (QMLE) of our model is considered, and its strong consistency and asymptotic normality are established under minimal assumptions on innovation. A new model diagnostic checking statistic is developed for model fitting adequacy. In addition, two methods for bubble tagging are proposed, one from the residual perspective and the other from the null-state perspective. Monte Carlo simulation studies are conducted to assess the performances of the QMLE and the two bubble tagging methods in finite samples. Finally, the usefulness of the model is illustrated by an empirical application to the monthly Hang Seng Index.

arXiv📈🤖
Bubble Modeling and Tagging: A Stochastic Nonlinear Autoregression Approach
By Yang, Li, Zhang

This study presents a framework for high-resolution mortality simulations tailored to insured and general populations. Due to the scarcity of detailed demographic-specific mortality data, we leverage Iterative Proportional Fitting (IPF) and Monte Carlo simulations to generate refined mortality tables that incorporate age, gender, smoker status, and regional distributions. This methodology enhances public health planning and actuarial analysis by providing enriched datasets for improved life expectancy projections and insurance product development.

arXiv📈🤖
Mortality simulations for insured and general populations
By Nalmpatian, Heumann

Geochemical data are compositional in nature and are subject to the problems typically associated with data that are restricted to the real non-negative number space with constant-sum constraint, that is, the simplex. Geochemistry can be considered a proxy for mineralogy, comprised of atomically ordered structures that define the placement and abundance of elements in the mineral lattice structure. Based on the innovative contributions of John Aitchison, who introduced the logratio transformation into compositional data analysis, this contribution provides a systematic workflow for assessing geochemical data in a simple and efficient way, such that significant geochemical (mineralogical) processes can be recognized and validated. This workflow, called GeoCoDA and presented here in the form of a tutorial, enables the recognition of processes from which models can be constructed based on the associations of elements that reflect mineralogy. Both the original compositional values and their transformation to logratios are considered. These models can reflect rock-forming processes, metamorphism, alteration and ore mineralization. Moreover, machine learning methods, both unsupervised and supervised, applied to an optimized set of subcompositions of the data, provide a systematic, accurate, efficient and defensible approach to geochemical data analysis. The workflow is illustrated on lithogeochemical data from exploration of the Star kimberlite, consisting of a series of eruptions with five recognized phases.

arXiv📈🤖
GeoCoDA: Recognizing and Validating Structural Processes in Geochemical Data
By

Sample selection is pervasive in applied economic studies. This paper develops semiparametric selection models that achieve point identification without relying on exclusion restrictions, an assumption long believed necessary for identification in semiparametric selection models. Our identification conditions require at least one continuously distributed covariate and certain nonlinearity in the selection process. We propose a two-step plug-in estimator that is root-n-consistent, asymptotically normal, and computationally straightforward (readily available in statistical software), allowing for heteroskedasticity. Our approach provides a middle ground between Lee (2009)'s nonparametric bounds and Honor\'e and Hu (2020)'s linear selection bounds, while ensuring point identification. Simulation evidence confirms its excellent finite-sample performance. We apply our method to estimate the racial and gender wage disparity using data from the US Current Population Survey. Our estimates tend to lie outside the Honor\'e and Hu bounds.

arXiv📈🤖
Point-Identifying Semiparametric Sample Selection Models with No Excluded Variable
By Kim, Lee

Biochemical discovery increasingly relies on classifying molecular structures when the consequences of different errors are highly asymmetric. In mutagenicity and carcinogenicity, misclassifying a harmful compound as benign can trigger substantial scientific, regulatory, and health risks, whereas false alarms primarily increase laboratory workload. Modern representations transform molecular graphs into persistence image tensors that preserve multiscale geometric and topological structure, yet existing tensor classifiers and deep tensor neural networks provide no finite-sample guarantees on type I error and often exhibit severe error inflation in practice. We develop the first Tensor Neyman-Pearson (Tensor-NP) classification framework that achieves finite-sample control of type I error while exploiting the multi-mode structure of tensor data. Under a tensor-normal mixture model, we derive the oracle NP discriminant, characterize its Tucker low-rank manifold geometry, and establish tensor-specific margin and conditional detection conditions enabling high-probability bounds on excess type II error. We further propose a Discriminant Tensor Iterative Projection estimator and a Tensor-NP Neural Classifier combining deep learning with Tensor-NP umbrella calibration, yielding the first distribution-free NP-valid methods for multiway data. Across four biochemical datasets, Tensor-NP classifiers maintain type I errors at prespecified levels while delivering competitive type II error performance, providing reliable tools for asymmetric-risk decisions with complex molecular tensors.

arXiv📈🤖
Tensor Neyman-Pearson Classification: Theory, Algorithms, and Error Control
By Liu, Chen, Han et al

arXiv:2302.07658v2 Announce Type: replace-cross Abstract: Monitoring the quality of statistical processes has been of great importance, mostly in industrial applications. Control charts are widely used for this purpose, but often lack the possibility to monitor survival outcomes. Recently, inspecting survival outcomes has become of interest, especially in medical settings where outcomes often depend on risk factors of patients. For this reason many new survival control charts have been devised and existing ones have been extended to incorporate survival outcomes. The R package success allows users to construct risk-adjusted control charts for survival data. Functions to determine control chart parameters are included, which can be used even without expert knowledge on the subject of control charts. The package allows to create static as well as interactive charts, which are built using ggplot2 (Wickham 2016) and plotly (Sievert 2020).

arXiv📈🤖
SUrvival Control Chart EStimation Software in R: the success package
By

arXiv:2410.05484v1 Announce Type: cross Abstract: Despite their success and widespread adoption, the opaque nature of deep neural networks (DNNs) continues to hinder trust, especially in critical applications. Current interpretability solutions often yield inconsistent or oversimplified explanations, or require model changes that compromise performance. In this work, we introduce TRACER, a novel method grounded in causal inference theory designed to estimate the causal dynamics underpinning DNN decisions without altering their architecture or compromising their performance. Our approach systematically intervenes on input features to observe how specific changes propagate through the network, affecting internal activations and final outputs. Based on this analysis, we determine the importance of individual features, and construct a high-level causal map by grouping functionally similar layers into cohesive causal nodes, providing a structured and interpretable view of how different parts of the network influence the decisions. TRACER further enhances explainability by generating counterfactuals that reveal possible model biases and offer contrastive explanations for misclassifications. Through comprehensive evaluations across diverse datasets, we demonstrate TRACER's effectiveness over existing methods and show its potential for creating highly compressed yet accurate models, illustrating its dual versatility in both understanding and optimizing DNNs.

arXiv📈🤖
Neural Networks Decoded: Targeted and Robust Analysis of Neural Network Decisions via Causal Explanations and Reasoning
By

Methods for population estimation and inference have evolved over the past decade to allow for the incorporation of spatial information when using capture-recapture study designs. Traditional approaches to specifying spatial capture-recapture (SCR) models often rely on an individual-based detection function that decays as a detection location is farther from an individual's activity center. Traditional SCR models are intuitive because they incorporate mechanisms of animal space use based on their assumptions about activity centers. We modify the SCR model to accommodate a wide range of space use patterns, including for those individuals that may exhibit traditional elliptical utilization distributions. Our approach uses underlying Gaussian processes to characterize the space use of individuals. This allows us to account for multimodal and other complex space use patterns that may arise due to movement. We refer to this class of models as geostatistical capture-recapture (GCR) models. We adapt a recursive computing strategy to fit GCR models to data in stages, some of which can be parallelized. This technique facilitates implementation and leverages modern multicore and distributed computing environments. We demonstrate the application of GCR models by analyzing both simulated data and a data set involving capture histories of snowshoe hares in central Colorado, USA.

arXiv📈🤖
Geostatistical capture-recapture models
By

Spatial trend estimation under potential heterogeneity is an important problem to extract spatial characteristics and hazards such as criminal activity. By focusing on quantiles, which provide substantial information on distributions compared with commonly used summary statistics such as means, it is often useful to estimate not only the average trend but also the high (low) risk trend additionally. In this paper, we propose a Bayesian quantile trend filtering method to estimate the non-stationary trend of quantiles on graphs and apply it to crime data in Tokyo between 2013 and 2017. By modeling multiple observation cases, we can estimate the potential heterogeneity of spatial crime trends over multiple years in the application. To induce locally adaptive Bayesian inference on trends, we introduce general shrinkage priors for graph differences. Introducing so-called shadow priors with multivariate distribution for local scale parameters and mixture representation of the asymmetric Laplace distribution, we provide a simple Gibbs sampling algorithm to generate posterior samples. The numerical performance of the proposed method is demonstrated through simulation studies.

arXiv📈🤖
Locally Adaptive Spatial Quantile Smoothing: Application to Monitoring Crime Density in Tokyo
By

Intrinsic Gaussian fields are used in many areas of statistics as models for spatial or spatio-temporal dependence, or as priors for latent variables. However, there are two major gaps in the literature: first, the number and flexibility of existing intrinsic models are very limited; second, theory, fast inference, and software are currently underdeveloped for intrinsic fields. We tackle these challenges by introducing the new flexible class of intrinsic Whittle--Mat\'ern Gaussian random fields obtained as the solution to a stochastic partial differential equation (SPDE). Exploiting sparsity resulting from finite-element approximations, we develop fast estimation and simulation methods for these models. We demonstrate the benefits of this intrinsic SPDE approach for the important task of kriging under extrapolation settings. Leveraging the connection of intrinsic fields to spatial extreme value processes, we translate our theory to an SPDE approach for Brown--Resnick processes for sparse modeling of spatial extreme events. This new paradigm paves the way for efficient inference in unprecedented dimensions. To demonstrate the wide applicability of our new methodology, we apply it in two very different areas: a longitudinal study of renal function data, and the modeling of marine heat waves using high-resolution sea surface temperature data.

arXiv📈🤖
Intrinsic Whittle--Mat\'ern fields and sparse spatial extremes
By Bolin, Braunsteins, Engelke et al

Bayesian multinomial logistic regression provides a principled, interpretable approach to multiclass classification, but posterior sampling becomes increasingly expensive as the model dimension grows. Prior work has studied scalability in the number of subjects and covariates; in contrast, this paper focuses on how computation changes as the number of outcome categories increases. To improve scalability in settings with numerous categories, we adapt a gamma-augmentation strategy to decouple category-specific coefficient updates, so that each category's coefficients can be updated conditional on a single auxiliary variable per subject, rather than on the full set of other categories' coefficients. Because the resulting coefficient conditionals are non-conjugate, we couple this augmentation with either adaptive Metropolis-Hastings or elliptical slice sampling. Through simulation and a real-data example, we compare effective sample size and effective sampling rate across several standard competitors. We find that the best-performing sampler depends on the dimension and imbalance regime, and that the proposed augmentation provides substantial speedups in scenarios with numerous categories.

arXiv📈🤖
Bayesian Multinomial Logistic Regression for Numerous Categories
By Fisher, McEvoy

Data privacy is important in the AI era, and differential privacy (DP) is one of the golden solutions. However, DP is typically applicable only if data have a bounded underlying distribution. We address this limitation by leveraging second-moment information from a small amount of public data. We propose Public-moment-guided Truncation (PMT), which transforms private data using the public second-moment matrix and applies a principled truncation whose radius depends only on non-private quantities: data dimension and sample size. This transformation yields a well-conditioned second-moment matrix, enabling its inversion with a significantly strengthened ability to resist the DP noise. Furthermore, we demonstrate the applicability of PMT by using penalized and generalized linear regressions. Specifically, we design new loss functions and algorithms, ensuring that solutions in the transformed space can be mapped back to the original domain. We have established improvements in the models' DP estimation through theoretical error bounds, robustness guarantees, and convergence results, attributing the gains to the conditioning effect of PMT. Experiments on synthetic and real datasets confirm that PMT substantially improves the accuracy and stability of DP models.

arXiv📈🤖
Differentially Private Truncation of Unbounded Data via Public Second Moments
By Cao, Bi, Zhang

Model selection is a central task in statistics, but standard methods are not robust in misspecified settings where the true data-generating process (DGP) is not in the set of candidate models. The key limitation is that existing methods -- including information criteria and Bayesian posteriors -- do not quantify uncertainty about how well each candidate model approximates the true DGP. In this paper, we introduce a novel approach to model selection based on modeling the likelihood values themselves. Specifically, given $K$ candidate models and $n$ observations, we view the $n\times K$ matrix of negative log-likelihood values as a random data matrix and observe that the expectation of each row is equal to the vector of Kullback--Leibler divergences between the $K$ models and the true DGP, up to an additive constant. We use a multivariate normal model to estimate and quantify uncertainty in this expectation, providing calibrated inferences for robust model selection under misspecification. The procedure is easy to compute, interpretable, and comes with theoretical guarantees, including consistency.

arXiv📈🤖
Robust model selection using likelihood as data
By Choi, Spencer, Miller

Quantifying uncertainty in future climate projections is hindered by the prohibitive computational cost of running physical climate models, which severely limits the availability of training data. We propose a data-efficient framework for emulating the internal variability of global climate fields, specifically designed to overcome these sample-size constraints. Inspired by copula modeling, our approach constructs a highly expressive joint distribution via a composite transformation to a multivariate standard normal space. We combine a nonparametric Bayesian transport map for spatial dependence modeling with flexible, spatially varying marginal models, essential for capturing non-Gaussian behavior and heavy-tailed extremes. These marginals are defined by a parametric model followed by a semi-parametric B-spline correction to capture complex distributional features. The marginal parameters are spatially smoothed using Gaussian-process priors with low-rank approximations, rendering the computational cost linear in the spatial dimension. When applied to global log-precipitation-rate fields at more than 50,000 grid locations, our stochastic surrogate achieves high fidelity, accurately quantifying the climate distribution's spatial dependence and marginal characteristics, including the tails. Using only 10 training samples, it outperforms a state-of-the-art competitor trained on 80 samples, effectively octupling the computational budget for climate research. We provide a Python implementation at https://github.com/jobrachem/ppptm .

arXiv📈🤖
Data-Efficient Generative Modeling of Non-Gaussian Global Climate Fields via Scalable Composite Transformations
By Brachem, Wiemann, Katzfuss

The study of causal effects in the presence of unmeasured spatially varying confounders has garnered increasing attention. However, a general framework for identifiability, which is critical for reliable causal inference from observational data, has yet to be advanced. In this paper, we study a linear model with various parametric model assumptions on the covariance structure between the unmeasured confounder and the exposure of interest. We establish identifiability of the treatment effect for many commonly 20 used spatial models for both discrete and continuous data, under mild conditions on the structure of observation locations and the exposure-confounder association. We also emphasize models or scenarios where identifiability may not hold, under which statistical inference should be conducted with caution.

arXiv📈🤖
Identifiability of Treatment Effects with Unobserved Spatially Varying Confounders
By Tang, Li, Li

Switchback experiments--alternating treatment and control over time--are widely used when unit-level randomization is infeasible, outcomes are aggregated, or user interference is unavoidable. In practice, experimentation must support fast product cycles, so teams often run studies for limited durations and make decisions with modest samples. At the same time, outcomes in these time-indexed settings exhibit serial dependence, seasonality, and occasional heavy-tailed shocks, and temporal interference (carryover or anticipation) can render standard asymptotics and naive randomization tests unreliable. In this paper, we develop a randomization-test framework that delivers finite-sample valid, distribution-free p-values for several null hypotheses of interest using only the known assignment mechanism, without parametric assumptions on the outcome process. For causal effects of interests, we impose two primitive conditions--non-anticipation and a finite carryover horizon m--and construct conditional randomization tests (CRTs) based on an ex ante pooling of design blocks into "sections," which yields a tractable conditional assignment law and ensures imputability of focal outcomes. We provide diagnostics for learning the carryover window and assessing non-anticipation, and we introduce studentized CRTs for a session-wise weak null that accommodates within-session seasonality with asymptotic validity. Power approximations under distributed-lag effects with AR(1) noise guide design and analysis choices, and simulations demonstrate favorable size and power relative to common alternatives. Our framework extends naturally to other time-indexed designs.

arXiv📈🤖
Randomization Tests in Switchback Experiments
By Liu, Zhong

We propose a new and interpretable class of high-dimensional tail dependence models based on latent linear factor structures. Specifically, extremal dependence of an observable vector is assumed to be driven by a lower-dimensional latent $K$-factor model, where $K \ll d$, thereby inducing an explicit form of dimension reduction. Geometrically, this is reflected in the support of the associated spectral dependence measure, whose intrinsic dimension is at most $K-1$. The loading structure may additionally exhibit sparsity, meaning that each component is influenced by only a small number of latent factors, which further enhances interpretability and scalability. Under mild structural assumptions, we establish identifiability of the model parameters and provide a constructive recovery procedure based on a margin-free tail pairwise dependence matrix, which also yields practical rank-based estimation methods. The framework combines naturally with marginal tail models and is particularly well suited to high-dimensional settings. We illustrate its applicability in a spatial wind energy application, where the latent factor structure enables tractable estimation of the risk that a large proportion of turbines simultaneously fall below their cut-in wind speed thresholds.

arXiv📈🤖
Dimension Reduction in Multivariate Extremes via Latent Linear Factor Models
By Boulin, B\"ucher

We re-examine the traditional Mean-Squared Error (MSE) forecasting paradigm by formally integrating an accuracy-timeliness trade-off: accuracy is defined by MSE (or target correlation) and timeliness by advancement (or phase excess). While MSE-optimized predictors are accurate in tracking levels, they sacrifice dynamic lead, causing them to lag behind changing targets. To address this, we introduce two `look-ahead' frameworks--Decoupling-from-Present (DFP) and Peak-Correlation-Shifting (PCS)--and provide closed-form solutions for their optimization. Notably, the classical MSE predictor is shown to be a special case within these frameworks. Dually, our methods achieve maximum advancement for any given accuracy level, so our approach reveals the complete efficient frontier of the accuracy-timeliness trade-off, whereas MSE represents only a single point. We also derive a universal upper bound on lead over MSE for any linear predictor under a consistency constraint and prove that our methods hit this ceiling. We validate this approach through applications in forecasting and real-time signal extraction, introducing a leading-indicator criterion and tailored linear benchmarks.

arXiv📈🤖
Forecasting on the Accuracy-Timeliness Frontier: Two Novel `Look Ahead' Predictors
By Wildi

Sensitivity and specificity evaluated at an optimal diagnostic cut-off are fundamental measures of classification accuracy when continuous biomarkers are used for disease diagnosis. Joint inference for these quantities is challenging because their estimators are evaluated at a common, data-driven threshold estimated from both diseased and healthy samples, inducing statistical dependence. Existing approaches are largely based on parametric assumptions or fully nonparametric procedures, which may be sensitive to model misspecification or lack efficiency in moderate samples. We propose a semiparametric framework for joint inference on sensitivity and specificity at the Youden-optimal cut-off under the density ratio model. Using maximum empirical likelihood, we derive estimators of the optimal threshold and the corresponding sensitivity and specificity, and establish their joint asymptotic normality. This leads to Wald-type and range-preserving logit-transformed confidence regions. Simulation studies show that the proposed method achieves accurate coverage with improved efficiency relative to existing parametric and nonparametric alternatives across a variety of distributional settings. An analysis of COVID-19 antibody data demonstrates the practical advantages of the proposed approach for diagnostic decision-making.

arXiv📈🤖
Semiparametric Joint Inference for Sensitivity and Specificity at the Youden-Optimal Cut-Off
By Liu, Tian, Wang et al

We consider hypothesis testing of binary causal queries using observational data. Since the mapping of causal models to the observational distribution that they induce is not one-to-one, in general, causal queries are often only partially identifiable. When binary statistical tests are used for testing partially-identifiable causal queries, their results do not translate in a straightforward manner to the causal hypothesis testing problem. We propose using ternary (three-outcome) statistical tests to test partially-identifiable causal queries. We establish testability requirements that ternary tests must satisfy in terms of uniform consistency and present equivalent topological conditions on the hypotheses. To leverage the existing toolbox of binary tests, we prove that obtaining ternary tests by combining binary tests is complete. Finally, we demonstrate how topological conditions serve as a guide to construct ternary tests for two concrete causal hypothesis testing problems, namely testing the instrumental variable (IV) inequalities and comparing treatment efficacy.

arXiv📈🤖
Testing Partially-Identifiable Causal Queries Using Ternary Tests
By Bhadane, Mooij, Boeken et al

Permutation procedures are common practice in hypothesis testing when distributional assumptions about the test statistic are not met or unknown. With only few permutations, empirical p-values lie on a coarse grid and may even be zero when the observed test statistic exceeds all permuted values. Such zero p-values are statistically invalid and hinder multiple testing correction. Parametric tail modeling with the Generalized Pareto Distribution (GPD) has been proposed to address this issue, but existing implementations can again yield zero p-values when the estimated shape parameter is negative and the fitted distribution has a finite upper bound. We introduce a method for accurate and zero-free p-value approximation in permutation testing, embedded in the permApprox workflow and R package. Building on GPD tail modeling, the method enforces a support constraint during parameter estimation to ensure valid extrapolation beyond the observed statistic, thereby strictly avoiding zero p-values. The workflow further integrates robust parameter estimation, data-driven threshold selection, and principled handling of hybrid p-values that are discrete in the bulk and continuous in the extreme tail. Extensive simulations using two-sample t-tests and Wilcoxon rank-sum tests show that permApprox produces accurate, robust, and zero-free p-value approximations across a wide range of sample and effect sizes. Applications to single-cell RNA-seq and microbiome data demonstrate its practical utility: permApprox yields smooth and interpretable p-value distributions even with few permutations. By resolving the zero-p-value problem while preserving accuracy and computational efficiency, permApprox enables reliable permutation-based inference in high-dimensional and computationally intensive settings.

arXiv📈🤖
permApprox: a general framework for accurate permutation p-value approximation
By Peschel, Boulesteix, Mutius et al

Improper priors are not allowed for the computation of the Bayesian evidence $Z=p({\bf y})$ (a.k.a., marginal likelihood), since in this case $Z$ is not completely specified due to an arbitrary constant involved in the computation. However, in this work, we remark that they can be employed in a specific type of model selection problem: when we have several (possibly infinite) models belonging to the same parametric family (i.e., for tuning parameters of a parametric model). However, the quantities involved in this type of selection cannot be considered as Bayesian evidences: we suggest to use the name ``fake evidences'' (or ``areas under the likelihood'' in the case of uniform improper priors). We also show that, in this model selection scenario, using a diffuse prior and increasing its scale parameter asymptotically to infinity, we cannot recover the value of the area under the likelihood, obtained with a uniform improper prior. We first discuss it from a general point of view. Then we provide, as an applicative example, all the details for Bayesian regression models with nonlinear bases, considering two cases: the use of a uniform improper prior and the use of a Gaussian prior, respectively. A numerical experiment is also provided confirming and checking all the previous statements.

arXiv📈🤖
A note on the area under the likelihood and the fake evidence for model selection
By Martino, Llorente

Statistical analysis of functional data is challenging due to their complex patterns, for which functional depth provides an effective means of reflecting their ordering structure. In this work, we investigate practical aspects of the recently proposed regularized projection depth (RPD), which induces a meaningful ordering of functional data while appropriately accommodating their complex shape features. Specifically, we examine the impact and choice of its tuning parameter, which regulates the degree of effective dimension reduction applied to the data, and propose a random projection-based approach for its efficient computation, supported by theoretical justification. Through comprehensive numerical studies, we explore a wide range of statistical applications of the RPD and demonstrate its particular usefulness in uncovering shape features in functional data analysis. This ability allows the RPD to outperform competing depth-based methods, especially in tasks such as functional outlier detection, classification, and two-sample hypothesis testing.

arXiv📈🤖
Projection depth for functional data: Practical issues, computation and applications
By Bo\v{c}inec, Nagy, Yeon

We are honoured to have our work read and discussed at such a thorough level by several experts. Words of appreciation and encouragement are gratefully received, while the many supplementary comments, thoughtful reminders, new perspectives and additional themes raised are warmly welcomed and deeply appreciated. Our thanks go also to JASA Editor Francisco Samaniego and his editorial helpers for organising this discussion. Space does not allow us answering all of the many worthwhile points raised by our discussants, but in the following we make an attempt to respond to what we perceive of as being the major issues. Our responses are organised by themes rather than by discussants. We shall refer to our two articles as `the FMA paper' (Hjort and Claeskens) and `the FIC paper' (Claeskens and Hjort).

arXiv📈🤖
Rejoinder to the discussants of the two JASA articles `Frequentist Model Averaging' and `The Focused Information Ctierion', by Nils Lid Hjort and Gerda Claeskens
By Hjort, Claeskens

Multi-armed bandit (MAB) processes constitute a foundational subclass of reinforcement learning problems and represent a central topic in statistical decision theory, but are limited to simultaneous adaptive allocation and sequential test, because of the absence of asymptotic theory under non-i.i.d sequence and sublinear information. To address this open challenge, we propose Urn Bandit (UNB) process to integrate the reinforcement mechanism of urn probabilistic models with MAB principles, ensuring almost sure convergence of resource allocation to optimal arms. We establish the joint functional central limit theorem (FCLT) for consistent estimators of expected rewards under non-i.i.d., non-sub-Gaussian and sublinear reward samples with pairwise correlations across arms. To overcome the limitations of existing methods that focus mainly on cumulative regret, we establish the asymptotic theory along with adaptive allocation that serves powerful sequential test, such as arms comparison, A/B testing, and policy valuation. Simulation studies and real data analysis demonstrate that UNB maintains statistical test performance of equal randomization (ER) design but obtain more average rewards like classical MAB processes.

arXiv📈🤖
Asymptotic Theory and Sequential Test for General Multi-Armed Bandit Process
By Yang, Yan, Jiang

Streaming data often exhibit heterogeneity due to heteroscedastic variances or inhomogeneous covariate effects. Online renewable quantile and expectile regression methods provide valuable tools for detecting such heteroscedasticity by combining current data with summary statistics from historical data. However, quantile regression can be computationally demanding because of the non-smooth check function. To address this, we propose a novel online renewable method based on expectile regression, which efficiently updates estimates using both current observations and historical summaries, thereby reducing storage requirements. By exploiting the smoothness of the expectile loss function, our approach achieves superior computational efficiency compared with existing online renewable methods for streaming data with heteroscedastic variances or inhomogeneous covariate effects. We establish the consistency and asymptotic normality of the proposed estimator under mild regularity conditions, demonstrating that it achieves the same statistical efficiency as oracle estimators based on full individual-level data. Numerical experiments and real-data applications demonstrate that our method performs comparably to the oracle estimator while maintaining high computational efficiency and minimal storage costs.

arXiv📈🤖
Renewable estimation in linear expectile regression models with streaming data sets
By Cao, Wanga, Hua

Corner kicks are an important event in soccer because they are often the result of strong attacking play and can be of keen interest to sports fans and bettors. Peng, Hu, and Swartz (2024, Computational Statistics) formulate the mixture feature of corner kick times caused by previous corner kicks, frame the commonly available corner kick data as right-censored event times, and explore patterns of corner kicks. This paper extends their modeling to accommodate the potential correlations between corner kicks by the same teams within the same games. We con- sider a frailty model for event times and apply the Monte Carlo Expec- tation Maximization (MCEM) algorithm to obtain the maximum like- lihood estimates for the model parameters. We compare the proposed model with the model in Peng, Hu, and Swartz (2024) using likelihood ratio tests. The 2019 Chinese Super League (CSL) data are employed throughout the paper for motivation and illustration.

arXiv📈🤖
Learning about Corner Kicks in Soccer by Analysis of Event Times Using a Frailty Model
By Isaacs, Hu, Peng et al