Sign test

Remarks

For matched pairs $(Y_i, Z_i)$ one applies the test to the differences $X_i = Y_i - Z_i$ with $m_0 = 0$. Continuity of the law gives $P(X_i = m_0) = 0$; with real data, observations equal to $m_0$ are usually dropped and $n$ is reduced.

Example: $n = 10$ observations, all above $m_0$, so $S = 10$. The one-sided p-value is $P(S \ge 10) = 2^{-10} \approx 0.001$ under $H_0$, and $H_0$ is rejected at level $0.05$.

Non-example: let $P(X_i < m_0) = P(X_i > m_0) = 1/4$ and $P(X_i = m_0) = 1/2$. Then $m_0$ is a median, but the law is not continuous, and $S$ has law $\mathrm{Bin}(n, 1/4)$ instead of $\mathrm{Bin}(n, 1/2)$.

The Wilcoxon signed-rank test also uses the sizes of the deviations $X_i - m_0$, at the cost of assuming symmetry.