Bootstrapping

Definition

Let $X\sim f(x|\theta )$ be a random variable with $\theta \in \Omega$, $\mathbf{X}=(X_1,X_2,\dots ,X_n)$ be a Random Sample of $X$, and $\hat{\theta }=\hat{\theta }(\mathbf{X})$ be a point estimator of $\theta$ Then, the bootstrap sample is $n$-dimensional sample vector ${\mathbf{X}}^{\ast }=(X_1^{\ast },X_2^{\ast },\dots ,X_n^{\ast })$ drawn with replacement from the vector of original samples $\mathbf{X}=(X_1,X_2,\dots ,X_n)$

Bootstrap Confidence Interval

Draw $B$ (large number) bootstrap samples ${\mathbf{X}}_1^{\ast },{\mathbf{X}}_2^{\ast },\dots ,{\mathbf{X}}_B^{\ast }$ and calculate confidence interval using the samples Calculate point estimators using the bootstrap samples ${\hat{\theta }}_j^{\ast }:=\hat{\theta }({\mathbf{X}}_j^{\ast })$, and define order statistics ${\hat{\theta }}_{(1)}^{\ast }\le {\hat{\theta }}_{(2)}^{\ast }\le \cdots \le {\hat{\theta }}_{(B)}^{\ast }$ for the estimators. Then, $100(1-\alpha )\%$ confidence bootstrap confidence interval for $\theta$ is $({\hat{\theta }}_{(m)}^{\ast },{\hat{\theta }}_{(B+1-m)}^{\ast })$, where $m=\lfloor \frac{\alpha }{2}b\rfloor$