> [!tldr] > Given a quantity $\theta$ to be estimated and observed data $\mathbf{x}$, the estimators of its $\alpha$-quantile $\hat{\theta}[\alpha]$ (such that $\mathbb{P}[\theta \le \hat{\theta}[\alpha]] =\alpha$ for arbitrary $\alpha$) can be treated as the approximate inverse cdf. > > For the actual inverse cdf $F^{-1}$, we have $f(\theta)=\left( \frac{ \partial F^{-1}(\alpha) }{ \partial \alpha } \right)^{-1}_{\alpha=F^{-1}(\theta)},$so replacing $F^{-1}$ with $\hat{\theta}[\alpha]$ we get the **confidence density** of $\theta$: $g_{\mathbf{x}}(t):= \left[ \frac{ \partial \hat{\theta}[\alpha] }{ \partial \alpha}\right]^{-1}_{\hat{\theta}[\alpha]=t} \approx \lim_{\epsilon \to 0} \frac{\mathbb{P}\big[t < \theta \le (t + \epsilon)\big]}{\epsilon}.$ This $\alpha$-quantile can be estimated using [[Bootstraps|bootstrap]] methods, e.g. for a bootstrapped sample $(\theta^{(1)},\dots, \theta^{(B)})$, we can define $\begin{align*} \hat{\theta}[\alpha]&:= \inf\{ t~|~\text{proportion of }(\theta ^{(b)}) \text{ that's} \le t \text{ is at least }\alpha\}\\ &=\inf\left\{ t ~|~ \sum_{b}\mathbf{1}\{ \theta^{(b) }\le t\}\ge\alpha B \right\}. \end{align*}$