Bootstrap Methods: Another Look at the Jackknife¶
Why this mattered¶
TBD
Abstract¶
We discuss the following problem: given a random sample $\mathbf{X} = (X_1, X_2, \cdots, X_n)$ from an unknown probability distribution $F$, estimate the sampling distribution of some prespecified random variable $R(\mathbf{X}, F)$, on the basis of the observed data $\mathbf{x}$. (Standard jackknife theory gives an approximate mean and variance in the case $R(\mathbf{X}, F) = \theta(\hat{F}) - \theta(F), \theta$ some parameter of interest.) A general method, called the "bootstrap," is introduced, and shown to work satisfactorily on a variety of estimation problems. The jackknife is shown to be a linear approximation method for the bootstrap. The exposition proceeds by a series of examples: variance of the sample median, error rates in a linear discriminant analysis, ratio estimation, estimating regression parameters, etc.
Related¶
- enables → CONFIDENCE LIMITS ON PHYLOGENIES: AN APPROACH USING THE BOOTSTRAP — Efron's bootstrap resampling method enabled Felsenstein's bootstrap confidence limits for estimating support on phylogenetic trees.
- enables → Regression Shrinkage and Selection Via the Lasso — Bootstrap resampling enabled empirical assessment of estimator stability, a key concern for evaluating lasso's variable-selection behavior.
- enables → Non‐parametric multivariate analyses of changes in community structure — Efron's bootstrap enables the 1993 community-structure paper by providing resampling-based significance tests for nonparametric multivariate change.
- cite ← CONFIDENCE LIMITS ON PHYLOGENIES: AN APPROACH USING THE BOOTSTRAP — Felsenstein adapts Efron's bootstrap resampling method to estimate confidence limits for phylogenetic trees.
- cite ← Regression Shrinkage and Selection Via the Lasso — The lasso uses bootstrap resampling ideas from Efron to assess prediction error and stability of fitted regression models.
- cite ← Non‐parametric multivariate analyses of changes in community structure — The community-structure paper uses bootstrap-style resampling logic to assess significance in non-parametric multivariate tests.