title: Directed Transfer Function: Unified Asymptotic Theory and Some of Its Implications.
abstract: To present a unified mathematical derivation of the frequency-dependent asymptotic behavior of the three main forms of directed transfer function (DTF).
  
  
  
  A synthesis of the results (proved in an extended Appendix) is followed by a series of Monte Carlo simulations of representative examples.
  
  
  
  DTF estimators are asymptotically normal when the true values are different from zero. Under the null hypothesis H0: DTF=0, the estimator is distributed as a linear combination of independent χ21variables.
  
  
  
  Null DTF rejection is shown to be achievable with identical performance irrespective of which DTF form is adopted.
  
  
  
  Together with recent allied partial directed coherence results, this paper rounds up connectivity inference tools for a class of frequency-domain connectivity estimators.
authors.forename: Luiz A
authors.initials: LA
authors.lastname: Baccala
authors.forename: Daniel Y
authors.initials: DY
authors.lastname: Takahashi
authors.forename: Koichi
authors.initials: K
authors.lastname: Sameshima
mesh_terms: D001921:Brain / Q000502:physiology*; D006801:Humans; D008959:Models, Neurological; D015233:Models, Statistical; D009434:Neural Pathways / Q000502:physiology*; D012815:Signal Processing, Computer-Assisted