Some Finite Sample Properties of the Sign Test

TL;DR

This paper investigates finite-sample properties of the sign test, demonstrating its unbiasedness under certain conditions and highlighting its over-rejection issue with correlated data, with implications for small-sample inference.

Contribution

It provides a novel proof of the sign test's unbiasedness with non-i.i.d. data and presents a counterexample illustrating over-rejection with correlated data.

Findings

Sign test is unbiased with independent, non-identically distributed data.

Counterexample shows over-rejection occurs with correlated data.

Proof introduces a novel graph-theoretic argument.

Abstract

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.