bootstrap resampling methods

This paper addresses the complexity and high pedagogical/practical barriers associated with conventional uncertainty quantification methods—such as standard errors, confidence intervals, and hypothesis tests—in statistical inference. To evaluate the potential of nonparametric bootstrap as a unified alternative, we conduct a large-scale simulation study rigorously comparing single bootstrap, double bootstrap, and classical methods across multiple dimensions: sample size, confidence level, data-generating mechanisms, and statistical functionals. Results demonstrate that the double bootstrap consistently achieves superior coverage accuracy, stability, and robustness—particularly under small-sample and non-normal conditions—outperforming both classical approaches and the single bootstrap. We thus establish the double bootstrap as a principled, parsimonious, and high-performance paradigm for uncertainty quantification, providing both theoretical justification and empirical evidence to support its adoption in statistical education and applied practice.

This study addresses the unreliable estimation of repeatability, between-laboratory, and reproducibility variance components under ISO 5725 standards when sample sizes are small or variance structures are extreme. To overcome this limitation, the authors propose a tailored Bootstrap resampling strategy adapted to a one-way random effects model. The approach refines point estimates by adjusting within-laboratory resampling and constructs confidence intervals via a two-stage resampling scheme integrated with bias-corrected and accelerated (BCa) techniques. Extensive simulations and validation using real data from ISO 5725-4 demonstrate that the proposed method substantially improves estimation accuracy and confidence interval coverage. It yields reliable, near-nominal or conservatively valid inferences for small- to moderate-sized experiments and clearly delineates optimal strategies across different practical scenarios.

This paper addresses the lack of generality and theoretical foundations in existing bootstrap hypothesis testing frameworks. We propose a unified bootstrap testing framework that accommodates both null-distribution-based resampling and diverse nonstandard bootstrap schemes. We first systematically characterize the exchangeability condition and statistical functional construction criteria, prove the local asymptotic equivalence of different resampling schemes in terms of statistical power, and identify the intrinsic mechanism behind the failure of the naive bootstrap. Leveraging empirical process theory and weak convergence analysis, we rigorously establish the asymptotic exactness and consistency of the test under fixed alternatives. An accompanying open-source R package, *BootstrapTests*, validates the theoretical properties in independence testing, linear regression coefficient testing, and copula model goodness-of-fit testing. Finite-sample simulations demonstrate that the proposed method significantly improves statistical power.

This study addresses the stability assessment of group-specific ranking patterns and nonparametric inference on population-mean ordinal relationships in multivariate survey/scoring data. We propose the first hierarchical bootstrap framework for ordinal hypothesis testing, which approximates the null distribution without distributional assumptions. We introduce the *non-containment index*—a robust, interpretable measure quantifying ranking stability across groups—and leverage it for outlier response detection and significance testing of inter-group ranking differences. The method integrates hierarchical resampling, nonparametric stability analysis, and resampling-based ordinal inference, unifying descriptive and inferential capabilities. Evaluated in AI fairness auditing and questionnaire analysis, it demonstrates high sensitivity and reliability. Our approach establishes a novel, assumption-free, robust, and interpretable statistical paradigm for ordinal data analysis.

Traditional $n$-out-of-$n$ bootstrap fails for inconsistent estimators—such as extremes, quantiles, and nonsmooth $M$-estimators—due to asymptotic non-normality. To address this, we propose an automated implementation framework for the $m$-out-of-$n$ bootstrap. Our key methodological contribution is the first systematic development of adaptive estimation procedures for both the scaling factor $ au_n$ and the optimal subsample size $m$, grounded in asymptotic theory for inconsistent estimation. We rigorously evaluate multiple $m$-selection strategies via extensive Monte Carlo simulations, assessing their finite-sample coverage accuracy for confidence intervals. Based on this framework, we develop the R package `moonboot`, enabling robust confidence interval construction for diverse inconsistent estimators. Empirical results demonstrate that our approach substantially improves actual coverage probability in small samples, while maintaining theoretical validity and practical usability.

This study addresses the challenge in nonlinear mixed-effects models where conventional methods often fail to adequately account for the hierarchical structure of between-subject and within-subject variability, leading to undercoverage of confidence intervals—particularly for variance components. To overcome this limitation, the authors propose a conditional nonparametric bootstrap (cNP) approach that innovatively integrates the conditional distribution of individual random effects estimated via the SAEM algorithm with residual resampling. This method preserves the original data’s sample size, covariate distribution, and hierarchical structure without requiring explicit stratification. Simulation studies implemented using the saemix package demonstrate that, across various designs and levels of residual variability, cNP substantially improves coverage probabilities compared to classical nonparametric and case bootstrap methods, while excelling in maintaining the integrity of the data’s inherent structure.

Traditional bootstrap and conformal prediction methods fail in time series settings due to violations of exchangeability and the absence of a unified framework that supports dependence-aware resampling and adaptive conformal calibration. This work proposes the first typed API integrating block, residual, sieve, and wild resampling schemes with adaptive conformal approaches such as EnbPI and ACI, enabling distribution-free uncertainty quantification. Leveraging compilation-based acceleration and streaming reductions, the method requires only O(B) additional memory, circumventing the O(Bn) tensor duplication typical of conventional implementations. Empirical results demonstrate that the approach substantially mitigates undercoverage under the i.i.d. assumption, with sieve resampling achieving coverage closest to the nominal level for short-memory linear processes, while running several times faster than the arch benchmark.

This study addresses the limitations of commonly reported point estimates—such as F1 scores—in text classification, which often lack reliable uncertainty quantification, particularly in settings involving small samples, rare classes, or nested data structures (e.g., texts nested within individuals). The authors systematically evaluate multiple confidence interval methods and propose a bootstrap-based F1 estimator augmented with pseudocount regularization. They further demonstrate that accurate inference in nested designs requires simultaneous adjustment of both effective sample size and degrees of freedom. Empirical results show that conventional Wald intervals suffer from undercoverage, whereas the recommended Agresti–Coull, Wilson, and hierarchical bootstrap methods substantially improve coverage accuracy, offering a more robust approach to uncertainty quantification in domains such as the social sciences.

This work proposes a “cheap studentized bootstrap” that achieves the same high-order coverage accuracy as the conventional studentized bootstrap while drastically reducing computational cost. Traditional studentized bootstrap methods require extensive resampling or analytical standard error calculations, rendering them computationally expensive. The key innovation lies in formally establishing, for the first time, the connection between studentized statistics and the t-distribution, revealing that the degrees of freedom in the t-distribution reflect the amount of resampling computation rather than the original sample size. Building on Edgeworth and Cornish–Fisher expansions together with limiting t-distribution theory, the authors construct a high-order accuracy framework that maintains rigorous theoretical guarantees with only a minimal number of Monte Carlo resamples.