Resampling & Confidence Intervals

Let X₁, …, Xₙ be i.i.d. from F. The empirical CDF F̂ₙ(x) = (1/n) Σᵢ 1[Xᵢ ≤ x] — the fraction of observations at or below x — converges to F uniformly, almost surely, as n → ∞. Not just pointwise: the worst-case gap over all x vanishes.

Let θ = T(F) be the parameter of interest, θ̂ = T(F̂ₙ) the plug-in estimator from the observed data, and θ̂* = T(F̂ₙ*) the same functional computed from a bootstrap resample. The claim: for smooth functionals T, the conditional distribution of θ̂* − θ̂ (given the data) mimics the sampling distribution of θ̂ − θ. Proof sketch: (1) by Glivenko-Cantelli, F̂ₙ → F uniformly in probability; (2) if T is Hadamard-differentiable (a functional analogue of differentiability), then T(F̂ₙ) → T(F) in probability by the continuous mapping theorem; (3) the bootstrap distribution of θ̂* tracks the sampling distribution of θ̂ because the bootstrap world treats F̂ₙ as the truth, and F̂ₙ ≈ F. Condition: T must be a continuous (Hadamard-differentiable) functional. The mean, median, quantiles, and variance are consistent; the maximum is not (non-smooth — it is sensitive to the tail of F).

z₀ measures how far the median of θ̂* is from θ̂ on the standard normal scale. If the bootstrap distribution is symmetric around θ̂, exactly half of the θ̂*_b are below θ̂, giving z₀ = Φ⁻¹(0.5) = 0. If z₀ > 0, the bootstrap median is below θ̂ (an upward bias correction is needed); if z₀ < 0, the bootstrap median is above θ̂.

The acceleration a measures the rate at which the standard error of θ̂ changes with the true parameter. It is estimated by the jackknife: θ̂₍₋ᵢ₎ is the estimate computed with the i-th observation left out, and θ̄₍·₎ = (1/n) Σᵢ θ̂₍₋ᵢ₎ is the jackknife mean. a = 0 when the SE is constant (e.g. for the mean); a ≠ 0 for asymmetric statistics (e.g. variance, correlation).

For a (1 − α) confidence interval, compute the adjusted quantile levels α₁ and α₂, where z_p = Φ⁻¹(p) is the standard normal quantile. The BCa CI is then [θ̂*₍α₁₎, θ̂*₍α₂₎] — quantiles of the bootstrap distribution read at the adjusted levels instead of the raw 2.5 % and 97.5 %.