Signmean

Signmean is a nonparametric statistic that summarizes the central tendency of a data set relative to a chosen baseline. For a sample X = {x1, x2, ..., xn} and a baseline b, the signmean is defined as M = (1/n) ∑ sign(xi − b), where sign(t) equals −1 if t < 0, 0 if t = 0, and 1 if t > 0. The baseline b can be a fixed reference value or an estimated location parameter such as a mean, median, or another robust statistic.

Interpretation and relation to other methods: M measures the balance of observations above versus below the baseline. It lies in the interval [−1, 1], with M = 0 suggesting a balanced or symmetric distribution around the baseline (in the absence of ties). If ties are rare, M reduces to P(X > b) − P(X < b). The signmean is closely related to the sign test, but provides a continuous summary rather than a binary count, and can be used with a chosen baseline to assess shifts from that reference.

Estimation and properties: For IID data and a fixed baseline, E[M] = P(X > b) − P(X < b). The variance is Var(M) ≈ [(1 − r) − (p+ − p−)^2] / n, where p+ = P(X > b), p− = P(X < b), and r = P(X = b). Ties reduce information. Inference can use asymptotic normal theory or bootstrap methods, especially when the baseline is data-driven.

Applications and variants: Signmean offers a robust, simple descriptor for detecting shifts from a baseline in robust statistics, quality control, and signal processing. Variants include using different baselines (e.g., sample median) or weighting signs to reflect confidence in individual observations.

See also: sign test, nonparametric location estimators, Wilcoxon signed-rank test.