Cramér–von Mises criterion

Cramér–von Mises test (two samples)

Let ${\displaystyle x_1,x_2,\dots ,x_N}$ and ${\displaystyle y_1,y_2,\dots ,y_M}$ be the observed values in the first and second sample respectively, in increasing order. Let ${\displaystyle r_1,r_2,\dots ,r_N}$ be the ranks of the xs in the combined sample, and let ${\displaystyle s_1,s_2,\dots ,s_M}$ be the ranks of the ys in the combined sample. Anderson shows that

${\displaystyle T=\frac{NM}{N+M}{\omega }^2=\frac{U}{NM(N+M)}-\frac{4MN-1}{6(M+N)}}$

where U is defined as

${\displaystyle U=N\sum _{i=1}^N(r_i-i{)}^2+M\sum _{j=1}^M(s_j-j{)}^2}$

If the value of T is larger than the tabulated values, the hypothesis that the two samples come from the same distribution can be rejected. (Some books give critical values for U, which is more convenient, as it avoids the need to compute T via the expression above. The conclusion will be the same.)

The above assumes there are no duplicates in the ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle r}$ sequences. So ${\displaystyle x_i}$ is unique, and its rank is ${\displaystyle i}$ in the sorted list ${\displaystyle x_1,\dots ,x_N}$. If there are duplicates, and ${\displaystyle x_i}$ through ${\displaystyle x_j}$ are a run of identical values in the sorted list, then one common approach is the midrank method: assign each duplicate a "rank" of ${\displaystyle (i+j)/2}$. In the above equations, in the expressions ${\displaystyle (r_i-i{)}^2}$ and ${\displaystyle (s_j-j{)}^2}$, duplicates can modify all four variables ${\displaystyle r_i}$, ${\displaystyle i}$, ${\displaystyle s_j}$, and ${\displaystyle j}$.