Let , , where ω is the frequency and is a mean-zero error term with variance . The rank is . The expected rank is given by where . In this Demonstration, . In the top panel, the dots show the simulated values when the normal distribution is used with and . The bottom panel shows the average empirical rank (points) based on 1000 simulations and expected rank (curve). The bottom panel demonstrates that the frequency in the original data can be determined using the ranks, provided that enough data is available.

Snapshot 1: with , the top panel shows the signal-to-noise ratio is large, but with 1000 simulations the bottom panel shows that average ranks converge to the expected ranks

Snapshot 2: the top panel shows how skewed the errors are when the centered exponential distribution is used, while the bottom panel illustrates the convergence of the ranks to the expectations

Snapshot 3: the Cauchy distribution produces such extreme outliers that the signal in the top panel is not apparent, but even still, the average ranks converge to the predicted expected values

Reference

[1] Y. Lai and A. I. McLeod, "Robust Estimation of Frequency in Semiparametric Harmonic Regression," working paper.