Bootstrap confidence interval

A bootstrap confidence interval uses the conditional resampling distribution of an estimator to choose its endpoints. Basic intervals invert bootstrap error quantiles; percentile intervals use quantiles of the resampled estimates themselves. Their asymptotic coverage requires conditions appropriate to the statistic and interval construction.

For $\theta ={\beta }_1{\beta }_2$, use $\hat{\theta }={\hat{\beta }}_1{\hat{\beta }}_2$. Under regular regression assumptions, the delta method gives standard error $\hat{s}=(d^{\top }\hat{V}d{)}^{1/2}$, where $d=({\hat{\beta }}_2,{\hat{\beta }}_1{)}^{\top }$ and $\hat{V}$ estimates the covariance of the regression estimate. Resample the observational units, refit, and compute both ${\hat{\theta }}^{\ast }$ and ${\hat{s}}^{\ast }$. A percentile bootstrap confidence interval uses the .025 and .975 quantiles of ${\hat{\theta }}^{\ast }$. A bootstrap-t confidence interval uses quantiles of $({\hat{\theta }}^{\ast }-\hat{\theta })/{\hat{s}}^{\ast }$ and reverses them when solving for $\theta$. If both true coefficients vanish, the first derivative is zero and the standard first-order justification is unavailable; product-specific nonregular inference is then needed.

For each bootstrap sample, form $T^{\ast }=({\hat{\theta }}^{\ast }-\hat{\theta })/{\hat{s}}^{\ast }$, using that sample's estimated standard error, possibly from a nested bootstrap. If $q_p$ are conditional quantiles of $T^{\ast }$ and $\hat{s}$ is the original estimated standard error, invert the studentized inequalities to obtain the displayed interval. The quantile order is reversed by subtraction. Its coverage is approximate and requires a nondegenerate statistic and suitable bootstrap regularity.

A percentile bootstrap confidence interval takes the lower and upper chosen quantiles of the conditional bootstrap distribution of the estimator itself. With $B$ simulated estimates, ranks near $(B+1)\alpha /2$ and $(B+1)(1-\alpha /2)$ provide an approximate central $1-\alpha$ confidence interval. Unlike a basic bootstrap confidence interval, the endpoints are not reflected about the observed estimate. Coverage requires suitable regularity and accurate bootstrap approximation; the ranks do not establish exact finite-sample coverage.

A basic bootstrap interval reflects quantiles of the bootstrap estimation error about the observed estimate. For the sample mean it uses $[\bar{X}-q_{1-\alpha /2}/\sqrt{n},\bar{X}-q_{\alpha /2}/\sqrt{n}]$, where $q$ is a conditional quantile of $\sqrt{n}({\bar{X}}^b-\bar{X})$.