Determine the sample size needed to access population asymmetry

Determine the sample size required for federated and centralized cumulant-based LiNGAM estimators to recover causal-order information from population asymmetry rather than from the finite-sample variance ladder.

Background

At the sample sizes examined in the paper, the row-sum rankings of third- and fourth-order cumulant methods are dominated by variance stratification, even though the population asymmetry that identifies true sources is theoretically present. Marginal standardization removes the variance ladder and causes the cumulant-based methods to perform near randomly or below the random baseline.

The authors therefore leave unresolved the amount of data needed for the population asymmetry to become statistically accessible and to dominate the scale signal in federated cumulant aggregation. This question concerns both the absolute sample size and its dependence on dimension, graph topology, noise distribution, and the cumulant order.