Directed Transfer Function: Unified Asymptotic Theory and Some of Its Implications

Objective: To present a unified mathematical derivation of the frequency-dependent asymptotic behavior of the three main forms of directed transfer function (DTF). Methods: A synthesis of the results (proved in an extended Appendix) is followed by a series of Monte Carlo simulations of representative examples. Results: DTF estimators are asymptotically normal when the true values are different from zero. Under the null hypothesis H
₀
: DTF=0, the estimator is distributed as a linear combination of independent X
₁
²
variables. Conclusions: Null DTF rejection is shown to be achievable with identical performance irrespective of which DTF form is adopted. Significance: Together with recent allied partial directed coherence results, this paper rounds up connectivity inference tools for a class of frequency-domain connectivity estimators.