Mann-Whitney U Test

Background

For skewed, heavy-tailed metrics (revenue per user, watch time), the Welch $t$-test's normality assumption is shaky. The Mann-Whitney U is the non-parametric alternative: it tests whether one sample tends to produce larger values than another, using only ranks. No mean, no variance, no normality assumption.

Problem statement

Implement mann_whitney_u(sample_a, sample_b) returning the U-statistic for sample A.

$$U_A \;=\; R_A \;-\; \frac{n_A(n_A + 1)}{2}$$

where $R_A$ is the sum of A's ranks in the pooled sample. Ties get average ranks (the standard convention).

Input

• sample_a, sample_b - array-likes of floats.

Output

• float - the U statistic for sample A.

Examples

Input: a=[1,2,3], b=[4,5,6]
Output: 0.0   # A is entirely below B

Constraints

• Both samples must be non-empty; ValueError otherwise.
• Average ranks for ties.

Notes

• U in [0, n_a * n_b]. $U_A + U_B = n_A n_B$, so $U_A = 0$ means A is entirely smaller, $U_A = n_A n_B$ means A is entirely larger.