Quantiles

Definition

Let $X$ be a Random Variable with cdf $F(x)$, and ${\xi }_p=F^{-1}(p)$ be a $p$-th quantile of $X$

Properties

Estimator of Quantile

Let $Y_1<\cdots <Y_n$ be order statistics, then we can define an estimator of quantile using the order statistic ${\xi }_p=Y_k$ where $k=\lfloor (n+1)p\rfloor$

and call the $Y_k$ a $p$-th sample quantile

Confidence Intervals of Quantiles

$P(Y_i<{\xi }_p<Y_j)=\sum _{k=i}^{j-1}\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}=\gamma$ where $i<\lfloor (n+1)p\rfloor <j$

Select $i,j$ satisfying the equation