Exact results and bounds for the mean squared error of percentile bootstraps (Q1965937)

Let \(X_j\), \(j=1,\dots, n\), be an i.i.d. random sample with continuous distribution function \(F\), \(F([a,b])=1\), let \(F^0\) be an estimate of \(F\) based on the sample, and let \(\hat F\) be a bootstrapped version of \(F^0\) obtained from a resample with \(F^0\) as the underlying distribution. The paper examines the performance of bootstraps in terms of mean squared error (MSE) and integrated mean squared error \(MISE={\mathbf E}\|F-\hat F\|^2\). Four types of bootstrap procedures are considered: (E) Efron's basic (naive); (R) Rubin's (with the masses of \(X_j\) in \(\hat F\) taken as Dirichlet distributed variables); (SE) smoothed Efron's; and (HSB) histospline smoothed Bayesian (combination of (R) and (SE)). It is shown that \[ \text{for (R)}\quad MISE= 2 (n+1)^{-1} \int_a^b F(x)(1-F(x)) dx, \] \[ \text{for (HSB)}\quad B - 2 (n+1)^{-1} \int_a^b F(x)dx \leq MISE \leq B \] \[ \text{where} B=((n+1)(n+2))^{-1} \left[2(b-a)/3+3n\int_a^b F(x)dx - 2(n-1)\int_a^b F(x)^2dx\right]. \] \[ \; \] Analogous results are obtained for (E) and (SE).