Cramer-von Mises Test

Definition

The Cramer-von Mises test is a non-parametric test for the equality of continuous, distribution functions.

The Cramer-von Mises test statistic for a given CDF $F$ is defined as $CV=n{\int }_{-\infty }^{\infty }[F_n(t)-F(t){]}^{2}dF(t)=\frac{1}{12n}+\sum _{i=1}^n{\left[\frac{2i-1}{2n}-F(t_i)\right]}^2$ where $F_n$ is the Empirical Distribution Function based on the i.i.d. random variables $X_i,i=1,2,\dots ,n$.