Bootstrap Methods in Statistical Inference

Bootstrap methods form a class of non‐parametric resampling techniques used to assess the variability and distributional properties of statistical estimators. By repeatedly drawing samples with replacement from an observed dataset, the bootstrap constructs an empirical approximation to the sampling distribution of quantities such as means, regression coefficients or more complex statistics. This approach circumvents analytical difficulties in deriving exact distributions, especially when sample sizes are moderate or models are nonlinear. Modern developments encompass parametric bootstrap, in which model‐based sampling is employed; the Bayesian bootstrap, which interprets weights on observations as random measures; and specialised variants addressing dependent data, complex survey designs or functional data. Bootstrap techniques enable bias correction, construction of confidence intervals, hypothesis testing and standard‐error estimation across diverse fields including ecology, genomics and machine learning. Their flexibility, minimal reliance on distributional assumptions and ease of implementation have made bootstrap methods integral to contemporary statistical inference, supporting robust decision‐making in research and applied contexts worldwide.

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Technical terms

Bootstrap method: A resampling technique that generates replicate datasets by sampling with replacement from the original data to approximate the sampling distribution of a statistic.

Resampling: The process of drawing repeated samples from observed data (with or without replacement) to assess statistical properties without relying solely on asymptotic theory.

Confidence interval: A range of values derived from data, via bootstrap or analytical methods, that is believed to contain the true parameter with a given probability.

Variance estimator: A statistic, often derived by bootstrap, that quantifies the variability of an estimator across hypothetical repeated samples.

Weighted Bayesian bootstrap: A variant assigning random weights to observations according to a Dirichlet process, yielding a posterior‐analogue distribution without specifying a likelihood function.

References

1. Bootstrap Methods for Canonical Correlation Analysis of Functional Data. Journal of Data Science and Intelligent Systems (2023).
2. A Bootstrap Variance Estimation Method for Multistage Sampling and Two-Phase Sampling When Poisson Sampling Is Used at the Second Phase. Stats (2022).
3. Estimating Neural Network’s Performance with Bootstrap: A Tutorial. Machine Learning and Knowledge Extraction (2021).
4. Weighted Bayesian bootstrap for scalable posterior distributions. Canadian Journal of Statistics (2020).