Introduction to Quantiles

We want to find a solution $t$ to the problem: $F(t)\ge p$ to answer questions like “for which income do I belong to the richest 25%?”

How can we calculate this value?

Let $F$ be a CDF. From Analysis we know that:

• If $F$ is continous and strictly monotonic then $F(x)=p$ has $F^{-1}(p)$ as solution
• If $F$ is only continous then the solution is a nonempty compact interval (more than one solution)
• If $F$ does not attain value $p$ then there is a jump above it

• Quantile
• Quantile Function
• Inversion Lemma

Convergence of quantiles and qq plots

Let $F$ be a CDF which is the limit of a sequence of ECDFS.

• If $F$ has exactly one p-quantile then the smallest and largest quantiles of the sequence of ECDFs converge to this p-quantile
• Every Accumulation Point of the smallest and largest quantiles of the sequence of CDFs is a p-quantile of $F$

Simple tests for distribution equality:

• qq-Plot
• pp-Plot