qq-Plot

When $F=G$ then their qq-plot is on the diagonal.

The deviation of the diagonal can yield qualitative information about the CDFs.

Normality Testing

Use this R code to test if given data sequence comes from a Normal Distribution:

mu = 2
sug = 13
 
x = rnorm(10000, mu, sig)
qqnorm(x)
abline(mu, sig)

See Shapiro-Wilk Test of Normality for a way to transform this rather subjective test into a statistical test.

Other Distributions

For other qq-plots use the R package car with qqPlot(x, dist="unif") as an example.

Compare equality of two distributions

Determine if data sequence from strictly monotonic CDF $F$:

$\left\{\left(Q_F^{-}\left(\frac{k-1/2}{m}\right),x_{(k)}\right)\mid k\in \{1,\dots ,m\}\right\}$ Determine if two data sequences from the same strictly monotonic CDF:

$\left\{\left(Q_{F_n(\cdot ;y)}^{-}\left(\frac{k-1/2}{m}\right),x_{(k)}\right)\mid k\in \{1,\dots ,m\}\right\}$

CDFs with plateau

We only have convergence for CDFs which are strictly monotonic. We thus need to use a pp-Plot which displays the two distribution functions against each other.