Das $\alpha$ Perzentil

Der Wert, für den gilt, dass $\alpha$ Prozent aller Datenpunkte kleiner sind und $1-\alpha$ Prozent der Werte größer sind.

Definition

Let $F$ be a CDF. Then $x_p$ is called $p$ -Quantile if

F\left(x_{p}-\right) \leq p \leq F\left(x_{p}\right)

$F$ has exactly one $p$ -quantile when there is at most one $t$ with $F(t)=p$
Similar definition when working with real probabilities $P((-\infty,x_p])\geq p$ and $P([x_p,\infty))\geq 1-p$

$x_{0.25},x_{0.5},x_{0.75}$ are called quartiles.

We can use quantiles to calculate

We use the Quantile Function to quickly calculate quantiles of a given CDF.

Quantiles of a data sequence can be best visualized using a Boxplot.

Empirical Quantiles

\begin{aligned} &Q_{F_{n}(\cdot ; x)}^{-}(p)=\min \left\{t \in \mathbb{R} \mid \sum_{i=1}^{n} 1_{(-\infty, t]}\left(x_{(i)}\right) \geq n p\right\}=x_{(\lceil n p\rceil)} \\ &Q_{F_{n}(\cdot ; x)}^{+}(p)=\max \left\{t \in \mathbb{R} \mid \sum_{i=1}^{n} 1_{(-\infty, t)}\left(x_{(i)}\right) \leq n p\right\}=x_{(\lfloor n p\rfloor+1)} \end{aligned}

Smallest p-Quantile

$Q_{F}^{-}(p):=\min \{t \in \mathbb{R} \mid F(t) \geq p\}$ > Proof

Largest p-Quantile

$Q_{F}^{+}(p):=\max \{t \in \mathbb{R} \mid F(t-) \leq p\}$

\begin{equation}\large x_{p}:= \begin{cases}\frac{x_{N \cdot p}+x_{N \cdot p+1}}{2} & \text { falls } N \cdot p \text { ganzzahlig } \\ x_{\lfloor N \cdot p+1\rfloor} & \text { falls } N \cdot p \text { nicht ganzzahlig }\end{cases} \end{equation}