  This paper contains two finite-sample results concerning the sign test.
First, we show that the sign-test is unbiased with independent, non-identically
distributed data for both one-sided and two-sided hypotheses. The proof for the
two-sided case is based on a novel argument that relates the derivatives of the
power function to a regular bipartite graph. Unbiasedness then follows from the
existence of perfect matchings on such graphs. Second, we provide a simple
theoretical counterexample to show that the sign test over-rejects when the
data exhibits correlation. Our results can be useful for understanding the
properties of approximate randomization tests in settings with few clusters.
