Statistics Applications
Emergent Dynamical Spatial Boundaries in Emergency Medical Services: A Navier-Stokes Framework from First Principles
Tatsuru Kikuchi
Emergency medical services (EMS) response times are critical determinants of patient survival, yet existing approaches to spatial coverage analysis rely on discrete distance buffers or ad-hoc geographic information system (GIS) isochrones without theoretical foundation. This paper derives continuous spatial boundaries for emergency response from first principles using fluid dynamics (Navier-Stokes equations), demonstrating that response effectiveness decays exponentially with time: $\tau(t) = \tau_0 \exp(-\kappa t)$, where $\tau_0$ is baseline effectiveness and $\kappa$ is the temporal decay rate. Using 10,000 simulated emergency incidents from the National Emergency Medical Services Information System (NEMSIS), I estimate decay parameters and calculate critical boundaries $d^*$ where response effectiveness falls below policy-relevant thresholds. The framework reveals substantial demographic heterogeneity: elderly populations (85+) experience 8.40-minute average response times versus 7.83 minutes for younger adults (18-44), with 33.6% of poor-access incidents affecting elderly populations despite representing 5.2% of the sample. Non-parametric kernel regression validation confirms exponential decay is appropriate (mean squared error 8-12 times smaller than parametric), while traditional difference-in-differences analysis validates treatment effect existence (DiD coefficient = -1.35 minutes, $p < 0.001$). The analysis identifies vulnerable populations–elderly, rural, and low-income communities–facing systematically longer response times, informing optimal EMS station placement and resource allocation to reduce health disparities.
Read the full abstract ▾
Statistics Applications
A generalisation of the signal-to-noise ratio using proper scoring rules
Jochen Bröcker and Eviatar Bach
A generalised concept of the signal-to-noise ratio (or equivalently the ratio of predictable components, or RPC) is provided, based on proper scoring rules. This definition is the natural generalisation of the classical RPC, yet it allows one to define and analyse the signal-to-noise properties of any type of forecast that is amenable to scoring, thus drastically widening the applicability of these concepts. The methodology is illustrated through numerical examples of ensemble forecasts, scored using the continuous ranked probability score (CRPS), and of probability forecasts of a binary event, scored using the logarithmic score. Numerical examples are carried out using both synthetic data with prescribed signal-to-noise ratios as well as seasonal ensemble hindcasts of the North Atlantic Oscillation (NAO) index. The latter have previously been interpreted as having a signal-to-noise "paradox", or anomalous signal-to-noise ratio, using the RPC statistic. For the synthetic data, the RPC statistic as well as the scoring rule-based ones agree regarding which data sets exhibit anomalous signal-to-noise ratios, but exhibit different variance, indicating different statistical properties. For the NAO data, on the other hand, the different statistics are more equivocal on whether the signal-to-noise ratio is anomalous.
Read the full abstract ▾
Machine Learning (Statistics) · Cross-listed from cs.LG
DiNo and RanBu: Lightweight Predictions from Shallow Random Forests
Tiago Mendonça dos Santos, Rafael Izbicki and Luís Gustavo Esteves
Random Forest ensembles are a strong baseline for tabular prediction tasks, but their reliance on hundreds of deep trees often results in high inference latency and
memory demands, limiting deployment in latency-sensitive or resource-constrained environments. We introduce DiNo (Distance with Nodes) and RanBu (Random Bushes), two shallow-forest methods that convert a small set of depth-limited trees into efficient, distance-weighted predictors. DiNo measures cophenetic distances via the most recent common ancestor of observation pairs, while RanBu applies kernel smoothing to Breiman's classical proximity measure. Both approaches operate entirely after forest training: no additional trees are grown, and tuning of the single bandwidth parameter $h$ requires only lightweight
matrix-vector operations. Across three synthetic
benchmarks and 25 public
datasets, RanBu matches or exceeds the accuracy of full-depth random forests-particularly in high-noise settings-while reducing training plus inference time by up to 95%. DiNo achieves the best bias-variance trade-off in low-noise regimes at a modest computational cost. Both methods extend directly to
quantile regression, maintaining accuracy with substantial speed gains. The implementation is available as an open-source R/C++ package at https://github.com/tiagomendonca/dirf. We focus on structured tabular random samples (i.i.d.), leaving extensions to other modalities for future work.
Read the full abstract ▾
Machine Learning (Statistics) · Cross-listed from cs.LG
Amortized Active Generation of Pareto Sets
Daniel M. Steinberg, Asiri Wijesinghe, Rafael Oliveira et al. (+3 more)
We introduce active generation of Pareto sets (A-GPS), a new framework for online discrete black-box multi-objective optimization (MOO). A-GPS learns a generative model of the Pareto set that supports a-posteriori conditioning on user preferences. The method employs a class probability
estimator (CPE) to predict non-dominance relations and to condition the generative model toward high-performing regions of the search space. We also show that this non-dominance CPE implicitly estimates the probability of hypervolume improvement (PHVI). To incorporate subjective trade-offs, A-GPS introduces preference direction vectors that encode user-specified preferences in objective space. At each iteration, the model is updated using both Pareto membership and
alignment with these preference directions, producing an amortized generative model capable of sampling across the Pareto front without retraining. The result is a simple yet powerful approach that achieves high-quality Pareto set approximations, avoids explicit hypervolume computation, and flexibly captures user preferences. Empirical results on synthetic
benchmarks and
protein design tasks demonstrate strong sample efficiency and effective preference incorporation.
Read the full abstract ▾
All authors Daniel M. Steinberg, Asiri Wijesinghe, Rafael Oliveira, Piotr Koniusz, Cheng Soon Ong, Edwin V. Bonilla
Statistical Methodology
Autocorrelation Test under Frequent Mean Shifts
Ziyang Liu, Ning Hao, Yue Selena Niu et al. (+2 more)
Testing for the presence of autocorrelation is a fundamental problem in time series analysis. Classical methods such as the Box-Pierce test rely on the assumption of stationarity, necessitating the removal of non-stationary components such as trends or shifts in the mean prior to application. However, this is not always practical, particularly when the mean structure is complex, such as being piecewise constant with frequent shifts. In this work, we propose a new inferential framework for autocorrelation in time series data under frequent mean shifts. In particular, we introduce a Shift-Immune Portmanteau (SIP) test that reliably tests for autocorrelation and is robust against mean shifts. We illustrate an application of our method to nanopore sequencing data.
Read the full abstract ▾
All authors Ziyang Liu, Ning Hao, Yue Selena Niu, Han Xiao, Hongxu Ding
Machine Learning (Statistics) · Cross-listed from cs.LG
From Information to Generative Exponent: Learning Rate Induces Phase Transitions in SGD
Konstantinos Christopher Tsiolis, Alireza Mousavi-Hosseini and Murat A. Erdogdu
To understand feature learning dynamics in neural networks, recent theoretical works have focused on gradient-based learning of Gaussian single-index models, where the label is a nonlinear function of a latent one-dimensional projection of the input. While the sample complexity of online SGD is determined by the information exponent of the link function, recent works improved this by performing multiple gradient steps on the same sample with different learning rates – yielding a non-correlational update rule – and instead are limited by the (potentially much smaller) generative exponent. However, this picture is only valid when these learning rates are sufficiently large. In this paper, we characterize the relationship between learning rate(s) and sample complexity for a broad class of gradient-based
algorithms that encapsulates both correlational and non-correlational updates. We demonstrate that, in certain cases, there is a phase transition from an "information exponent regime" with small learning rate to a "generative exponent regime" with large learning rate. Our framework covers prior analyses of one-pass SGD and SGD with batch reuse, while also introducing a new layer-wise training
algorithm that leverages a two-timescales approach (via different learning rates for each layer) to go beyond correlational queries without reusing samples or modifying the loss from squared error. Our theoretical study demonstrates that the choice of learning rate is as important as the design of the
algorithm in achieving statistical and computational efficiency.
Read the full abstract ▾
Statistical Computation · Cross-listed from econ.EM
SLIM: Stochastic Learning and Inference in Nonlinear GMM Models
Xiaohong Chen, Min Seong Kim, Sokbae Lee et al. (+3 more)
We propose SLIM (
Stochastic Learning and Inference in overidentified Models), a new scalable
stochastic approximation
algorithm for nonlinear GMMs with many nonseparable moments and parameters. SLIM is shown to be consistent and asymptotically normal without a consistent initial
estimator or a globally convex GMM criterion. Random-scaling and plug-in based confidence set constructions remain valid when computational uncertainty can be non-negligible relative to sampling uncertainty. An optional second-order refinement achieves full-sample GMM efficiency. We develop $J$-tests for SLIM. The theoretical results are extended to models with clustered data of unequal cluster sizes. Monte Carlo experiments for a nonlinear demand system with 576 moments and 380 parameters demonstrate computational gains and scalability to one million observations. A LinkedIn application with 4.8 million college graduates illustrates clustered estimation and inference for first employment destinations.
Read the full abstract ▾
All authors Xiaohong Chen, Min Seong Kim, Sokbae Lee, Myung Hwan Seo, Myunghyun Song, Ling Zhong
Statistics Theory
Ayoub El Hanchi, Murat Erdogdu and Chris Maddison
What property of the data distribution determines the excess risk of principal component analysis? In this paper, we provide a precise answer to this question. We establish a central limit
theorem for the error of the principal subspace estimated by PCA, and derive the asymptotic distribution of its excess risk under the reconstruction loss. We obtain a non-asymptotic upper bound on the excess risk of PCA that recovers, in the large sample limit, our asymptotic characterization. Underlying our contributions is the following result: we prove that the negative block Rayleigh quotient, defined on the Grassmannian, is generalized self-concordant along geodesics emanating from its minimizer of maximum rotation less than $\pi/4$.
Read the full abstract ▾
Machine Learning (Statistics) · Cross-listed from q-bio.NC
Scalable inference of functional neural connectivity at submillisecond timescales
Arina Medvedeva, Edoardo Balzani, Alex H Williams et al. (+1 more)
The Poisson Generalized Linear Model (GLM) is a foundational tool for analyzing neural spike train data. However, standard implementations rely on discretizing spike times into binned count data, limiting temporal resolution and scalability. Here, we develop Monte Carlo (MC) methods and
polynomial approximations (PA) to the continuous-time analog of these models, and show them to be advantageous over their discrete-time counterparts. Further, we propose using a set of exponentially scaled Laguerre
polynomials as an orthogonal temporal basis, which improves filter identification and yields closed-form integral solutions under the
polynomial approximation. Applied to both synthetic and real spike-time data from rodent hippocampus, our methods demonstrate superior accuracy and scalability compared to traditional binned GLMs, enabling functional connectivity inference in large-scale neural recordings that are temporally precise on the order of synaptic dynamical timescales and in agreement with known anatomical properties of hippocampal subregions. We provide open-source implementations of both MC and PA
estimators, optimized for GPU acceleration, to facilitate adoption in the neuroscience community.
Read the full abstract ▾
All authors Arina Medvedeva, Edoardo Balzani, Alex H Williams, Stephen L Keeley
Machine Learning (Statistics)
A Spectral Framework for Graph Neural Operators: Convergence Guarantees and Tradeoffs
Roxanne Holden and Luana Ruiz
Graphons, as limits of graph sequences, provide an operator-theoretic framework for analyzing the asymptotic behavior of graph neural operators. Spectral convergence of sampled graphs to graphons induces convergence of the corresponding neural operators, enabling transferability analyses of graph neural networks (GNNs). This paper develops a unified spectral framework that brings together convergence results under different assumptions on the underlying graphon, including no regularity, global Lipschitz continuity, and piecewise-Lipschitz continuity. The framework places these results in a common operator setting, enabling direct comparison of their assumptions, convergence rates, and tradeoffs. We further illustrate the empirical tightness of these rates on synthetic and real-world graphs.
Read the full abstract ▾
Statistical Computation
To MCMC or not to MCMC: Evaluating non-MCMC methods for Bayesian penalized regression
Florian D. van Leeuwen and Sara van Erp
Markov Chain Monte Carlo (MCMC) sampling is computationally expensive, especially for complex models. Alternative methods make simplifying assumptions about the posterior to reduce computational burden, but their impact on predictive performance remains unclear. This paper compares MCMC and non-MCMC methods for high-dimensional penalized regression, examining when computational shortcuts are justified for prediction tasks. We conduct a comprehensive simulation study using high-dimensional tabular data, then validate findings with empirical
datasets featuring both continuous and binary outcomes. An in-depth analysis of one
dataset provides a step-by-step tutorial implementing various
algorithms in R. Our results show that mean-field variational inference consistently performs comparably to MCMC methods. In simulations, mean-field VI exhibited 3-90% higher MSE across scenarios while reducing runtime by 7-30x compared to
Hamiltonian Monte Carlo. Empirical
datasets revealed dramatic speed-ups (100-400x) in some cases with similar or superior predictive performance. However, performance varied: some cases showed over 100x MSE increases with only 30x speed-ups, highlighting the context-dependent nature of these trade-offs.
Read the full abstract ▾
Statistical Methodology
Bayesian analysis of flexible Heckman selection models using Hamiltonian Monte Carlo
Heeju Lim, Victor E. Lachos and Victor H. Lachos
The Heckman selection model is widely used in econometric analysis and other social sciences to address sample selection bias in data modeling. A common assumption in Heckman selection models is that the error terms follow an independent bivariate normal distribution. However, real-world data often deviates from this assumption, exhibiting heavy-tailed behavior, which can lead to inconsistent estimates if not properly addressed. In this paper, we propose a
Bayesian analysis of Heckman selection models that replace the Gaussian assumption with well-known members of the class of scale mixture of normal distributions, such as the Student's-t and contaminated normal distributions. For these complex structures, Stan's default No-U-Turn sampler is utilized to obtain posterior simulations. Through extensive simulation studies, we compare the performance of the Heckman selection models with normal, Student's-t and contaminated normal distributions. We also demonstrate the broad applicability of this methodology by applying it to medical care and labor supply data. The proposed
algorithms are implemented in the R package HeckmanStan.
Read the full abstract ▾
Statistical Methodology
Handling Missing Responses under Cluster Dependence with Applications to Language Model Evaluation
Zhenghao Zeng, David Arbour, Avi Feller et al. (+3 more)
Human annotations play a crucial role in evaluating the performance of GenAI models. Two common challenges in practice, however, are missing annotations (the response variable of interest) and cluster dependence among human-AI interactions (e.g., questions asked by the same user may be highly correlated). Reliable inference must address both these issues to achieve unbiased estimation and appropriately quantify uncertainty when estimating average scores from human annotations. In this paper, we analyze the doubly robust
estimator, a widely used method in missing data analysis and causal inference, applied to this setting and establish novel theoretical properties under cluster dependence. We further illustrate our findings through simulations and a real-world conversation quality
dataset. Our theoretical and empirical results underscore the importance of incorporating cluster dependence in missing response problems to perform valid statistical inference.
Read the full abstract ▾
All authors Zhenghao Zeng, David Arbour, Avi Feller, Ishita Dasgupta, Atanu R Sinha, Edward H. Kennedy
Machine Learning (Statistics)
Kernel Learning with Adversarial Features: Numerical Efficiency and Adaptive Regularization
Antônio H. Ribeiro, David Vävinggren, Dave Zachariah et al. (+2 more)
Adversarial training has emerged as a key technique to enhance model robustness against
adversarial input perturbations. Many of the existing methods rely on computationally expensive min-max problems that limit their application in practice. We propose a novel formulation of
adversarial training in reproducing kernel
Hilbert spaces, shifting from input to feature-space perturbations. This reformulation enables the exact solution of inner maximization and efficient optimization. It also provides a regularized
estimator that naturally adapts to the noise level and the smoothness of the underlying function. We establish conditions under which the feature-perturbed formulation is a relaxation of the original problem and propose an efficient optimization
algorithm based on iterative kernel ridge regression. We provide generalization bounds that help to understand the properties of the method. We also extend the formulation to multiple kernel learning. Empirical evaluation shows good performance in both clean and
adversarial settings.
Read the full abstract ▾
All authors Antônio H. Ribeiro, David Vävinggren, Dave Zachariah, Thomas B. Schön, Francis Bach
Machine Learning (Statistics) · Cross-listed from cs.LG
MOBO-OSD: Batch Multi-Objective Bayesian Optimization via Orthogonal Search Directions
Lam Ngo, Huong Ha, Jeffrey Chan et al. (+1 more)
Bayesian Optimization (BO) is a powerful tool for optimizing expensive black-box objective functions. While extensive research has been conducted on the single-objective optimization problem, the multi-objective optimization problem remains challenging. In this paper, we propose MOBO-OSD, a multi-objective
Bayesian Optimization
algorithm designed to generate a diverse set of Pareto optimal solutions by solving multiple constrained optimization problems, referred to as MOBO-OSD subproblems, along orthogonal search directions (OSDs) defined with respect to an approximated convex hull of individual objective minima. By employing a well-distributed set of OSDs, MOBO-OSD ensures broad coverage of the objective space, enhancing both solution diversity and hypervolume performance. To further improve the density of the set of Pareto optimal candidate solutions without requiring an excessive number of subproblems, we leverage a Pareto Front Estimation technique to generate additional solutions in the neighborhood of existing solutions. Additionally, MOBO-OSD supports batch optimization, enabling parallel function evaluations to accelerate the optimization process when resources are available. Through extensive experiments and analysis on a variety of synthetic and real-world
benchmark functions with two to six objectives, we demonstrate that MOBO-OSD consistently outperforms the state-of-the-art
algorithms. Our code implementation can be found at https://github.com/LamNgo1/mobo-osd.
Read the full abstract ▾
All authors Lam Ngo, Huong Ha, Jeffrey Chan, Hongyu Zhang
Machine Learning (Statistics)
Exponential Convergence Guarantees for Iterative Markovian Fitting
Marta Gentiloni Silveri, Giovanni Conforti and Alain Durmus
The Schrödinger Bridge (SB) problem has become a fundamental tool in computational optimal transport and generative modeling. To address this problem, ideal methods such as Iterative Proportional Fitting and Iterative Markovian Fitting (IMF) have been proposed-alongside practical approximations like
Diffusion Schrödinger Bridge and its Matching (DSBM) variant. While previous work have established asymptotic convergence guarantees for IMF, a quantitative, non-asymptotic understanding remains unknown. In this paper, we provide the first non-asymptotic exponential convergence guarantees for IMF under mild structural assumptions on the reference measure and marginal distributions, assuming a sufficiently large time horizon. Our results encompass two key regimes: one where the marginals are log-concave, and another where they are weakly log-concave. The analysis relies on new contraction results for the Markovian projection operator and paves the way to theoretical guarantees for DSBM.
Read the full abstract ▾
Machine Learning (Statistics) · Cross-listed from cs.RO
The Reality Gap in Robotics: Challenges, Solutions, and Best Practices
Elie Aljalbout, Jiaxu Xing, Angel Romero et al. (+9 more)
Machine learning has facilitated significant advancements across various robotics domains, including
navigation, locomotion, and manipulation. Many such achievements have been driven by the extensive use of simulation as a critical tool for training and testing robotic systems prior to their deployment in real-world environments. However, simulations consist of abstractions and approximations that inevitably introduce discrepancies between simulated and real environments, known as the reality gap. These discrepancies significantly hinder the successful transfer of systems from simulation to the real world. Closing this gap remains one of the most pressing challenges in robotics. Recent advances in sim-to-real transfer have demonstrated promising results across various platforms, including locomotion,
navigation, and manipulation. By leveraging techniques such as domain randomization, real-to-sim transfer, state and action abstractions, and sim-real co-training, many works have overcome the reality gap. However, challenges persist, and a deeper understanding of the reality gap's root causes and solutions is necessary. In this survey, we present a comprehensive overview of the sim-to-real landscape, highlighting the causes, solutions, and evaluation metrics for the reality gap and sim-to-real transfer.
Read the full abstract ▾
All authors Elie Aljalbout, Jiaxu Xing, Angel Romero, Iretiayo Akinola, Caelan Reed Garrett, Eric Heiden, Abhishek Gupta, Tucker Hermans, Yashraj Narang, Dieter Fox, Davide Scaramuzza, Fabio Ramos
Statistics Theory
Incomplete U-Statistics of Equireplicate Designs: Berry-Esseen Bound and Efficient Construction
Cesare Miglioli and Jordan Awan
U-statistics are a fundamental class of
estimators that generalize the sample mean and underpin much of nonparametric statistics. Although extensively studied in both statistics and probability, key challenges remain: their high computational cost - addressed partly through incomplete U-statistics - and their non-standard asymptotic behavior in the degenerate case, which typically requires resampling methods for hypothesis testing. This paper presents a novel perspective on U-statistics, grounded in hypergraph theory and combinatorial designs. Our approach bypasses the traditional Hoeffding decomposition, the main analytical tool in this literature but one that is highly sensitive to degeneracy. By characterizing the dependence structure of a U-statistic, we derive a Berry-Esseen bound valid for incomplete U-statistics of deterministic designs, yielding conditions under which Gaussian limiting distributions can be established even in degenerate cases and when the order diverges. We also introduce efficient
algorithms to construct incomplete U-statistics based on equireplicate designs, a subclass of deterministic designs that, in certain cases, achieve minimum variance. Beyond its theoretical contributions, our framework provides a systematic way to construct permutation-free counterparts to tests based on degenerate U-statistics, as demonstrated in experiments with kernel-based tests using the Maximum Mean Discrepancy and the
Hilbert-Schmidt Independence Criterion.
Read the full abstract ▾
Statistics Theory
Bayesian Prediction under Moment Conditioning
Nicholas G. Polson and Daniel Zantedeschi
Moment restrictions specify a class of laws, not a predictive model. We obtain one by conditioning an independent sample from a reference law on its empirical moments, and define prediction as the law of a fixed block selected from that conditioned ensemble. On a finite partition this law is an exact mixture over empirical types. Under exact feasibility and
lattice regularity, the mixing law has a Gaussian limit on the feasible tangent space, governed by the reduced Hessian, and the selected block approaches independent sampling from the Kullback-Leibler projection. A separate finite-sample bound gives the same product limit for general real-valued restrictions without
lattice assumptions. Refinement recovers the projection on the original sample space. Parameterizing the projected family produces a predictive product criterion with a local inverse-covariance expansion, connecting the construction to generalized method of moments.
Read the full abstract ▾
Statistical Methodology
A comparison of methods for designing hybrid type 2 cluster-randomized trials with continuous effectiveness and implementation endpoints
Melody Owen, Fan Li, Ruyi Liu et al. (+1 more)
Hybrid type 2 studies are gaining popularity for their ability to assess both implementation and health outcomes as co-primary endpoints. Often conducted as cluster-randomized trials (CRTs), five design methods can validly power these studies: p-value adjustment methods, combined outcomes approach, single weighted 1-DF test, disjunctive 2-DF test, and conjunctive test. We compared these methods theoretically and numerically. Theoretical comparisons of power equations allowed us to identify when one method had more or less power than another globally. We showed that p-value adjustment methods are always less powerful than both the combined outcomes approach and the single 1-DF test, and identified conditions where the disjunctive 2-DF test is less powerful than the single 1-DF test. To further investigate when power advantages shift, we conducted a large-scale numerical study using our novel crt2power R package, which calculates power or sample size for CRTs with two continuous co-primary endpoints using these methods. Across 45,000 input scenarios, we found specific patterns: when treatment effects are unequal, the disjunctive 2-DF test tends to be most powerful; when treatment effects are equal, the single 1-DF test tends to dominate. Together, these comparisons offer practical guidance for powering hybrid type 2 studies.
Read the full abstract ▾
All authors Melody Owen, Fan Li, Ruyi Liu, Donna Spiegelman
Other Statistics · Cross-listed from cs.HC
Optimizing Feature Ordering in Radar Charts for Multi-Profile Comparison
Albert Dorador
Radar charts are widely used to visualize multivariate data and compare multiple profiles across features. However, the visual clarity of radar charts can be severely compromised when feature values alternate drastically in magnitude around the circle, causing areas to collapse, which misrepresents relative differences. In the present work we introduce a permutation optimization strategy that reorders features to minimize polygon “spikiness” across multiple profiles simultaneously. The method is combinatorial (exhaustive search) for moderate numbers of features and uses a lexicographic minimax criterion that first considers overall smoothness (mean jump) and then the largest single jump as a tie-breaker. This preserves more global information and produces visually balanced arrangements. We discuss complexity, practical bounds, and relations to existing approaches that either change the visualization (e.g., OrigamiPlot) or learn orderings (e.g., Versatile Ordering Network). An example with two profiles and $p=6$ features (before/after ordering) illustrates the qualitative improvement. Keywords: data visualization, radar charts, combinatorial optimization, minimax optimization, feature ordering
Read the full abstract ▾
Statistics Theory
Testing Imprecise Hypotheses
Lucas Kania, Tudor Manole, Larry Wasserman et al. (+1 more)
Many scientific applications involve testing theories that are only partially specified. This task often amounts to testing the goodness-of-fit of a candidate distribution while allowing for reasonable deviations from it. The tolerant testing framework provides a systematic way of constructing such tests. Rather than testing the simple null hypothesis that data was drawn from a candidate distribution, a tolerant test assesses whether the data is consistent with any distribution that lies within a given neighborhood of the candidate. As this neighborhood grows, the tolerance to misspecification increases, while the power of the test decreases. In this work, we characterize the information-theoretic trade-off between the size of the neighborhood and the power of the test, in several canonical models. On the one hand, we characterize the optimal trade-off for tolerant testing in the Gaussian sequence model, under deviations measured in both smooth and non-smooth norms. On the other hand, we study nonparametric analogues of this problem in smooth regression and density models. Along the way, we establish the sub-optimality of the classical chi-squared statistic for tolerant testing, and study simple alternative hypothesis tests.
Read the full abstract ▾
All authors Lucas Kania, Tudor Manole, Larry Wasserman, Sivaraman Balakrishnan