bootstrap confidence intervals

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.

Conventional methods for constructing confidence intervals for the mean under small samples suffer from inherent limitations: the bootstrap-t method yields overly wide, unstable, or even infinite intervals, while the BCₐ method exhibits severe under-coverage. Method: This paper proposes a novel Beta(1/2, 3/2)-weighted bootstrap-t approach, integrating Bayesian bootstrap principles with studentized statistics and implemented via nonparametric Monte Carlo simulation to achieve second-order accuracy. Contribution/Results: Theoretically and empirically, the proposed method eliminates the risk of infinite intervals and delivers coverage probabilities markedly closer to the nominal level across diverse small-sample settings. Its average interval length is significantly shorter than that of the standard bootstrap-t, and it consistently outperforms both BCₐ and polynomial bootstrap-t methods—achieving superior balance between coverage accuracy and statistical efficiency.

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.

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 Bayesian inference for low-dimensional target parameters in semiparametric models, particularly under the presence of complex nuisance components that may compromise frequentist properties. To this end, we construct posterior distributions by integrating estimating function methods with nonparametric Bayesian techniques—such as Dirichlet processes and Bayesian bootstrap—under conditions weaker than the classical stochastic equicontinuity assumption. We establish asymptotic normality and consistency of the resulting posterior, rigorously identifying the key assumptions required to guarantee desirable frequentist behavior. The theoretical analysis systematically elucidates how relaxing these assumptions affects inferential performance. Extensive simulations corroborate the effectiveness of the proposed methodology, demonstrating its robustness and accuracy in practical settings.

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 methods are computationally expensive and lack theoretical guarantees in semi-parametric and machine learning settings, undermining inference reliability. This work proposes V-fold jackknife as an efficient alternative: by refitting the estimator only V times on leave-one-fold-out samples, it quantifies uncertainty directly through the empirical variance of jackknife pseudovalues, bypassing explicit influence function calculations. For the first time, the authors establish a t-distribution limit theory for Studentized V-fold jackknife statistics with fixed V and extend it to generalized asymptotically linear estimators, enabling scale-invariant inference without requiring knowledge of convergence rates. The method demonstrates robust performance across diverse applications—including average treatment effects, Kaplan–Meier survival curves, and highly adaptive Lasso dose–response curves—and remains reliable even when influence functions fail to exist or are difficult to estimate.

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 work addresses the challenge of constructing confidence intervals with reliable coverage properties in the absence of exact group symmetry. It proposes Data Augmentation Bootstrap (DAB), a novel framework that integrates data augmentation into statistical inference by leveraging approximate invariance under transformations. DAB unifies existing approaches—such as bootstrap, conformal prediction, and SymmPI—as special cases through a common formulation based on Kolmogorov distance to quantify approximate invariance. By matching conditional means and variances under Gaussian universality, the method provides finite-sample and asymptotic coverage guarantees without requiring explicit group structure assumptions. Empirical evaluations across image, language, and scientific datasets demonstrate that DAB substantially improves the coverage performance of diverse inference methods.