Well, it is rather complicated! And I am quite new to this whole idea, as well. I had heard of the theory that Pilate started in AD 19 or so, but had never analyzed it, which is what I am beginning to do now.
Remember that our extant text of Josephus tells us that Gratus governed for eleven years and Pilate for ten years. If those two lines are correct, then Gratus started in around AD 15, after the death of Augustus, and that means that Pilate started in around AD 26. The lack of explicit dates in Josephus for other events, especially the death of Germanicus and the Mundus and Paulina debacle, allows us to assume that Josephus has narrated those events out of order, since both are narrated in connection with Pilate rather than with Gratus, whereas Tacitus dates them both to around AD 19. What Eisler and Schwartz point out is that, without those two chronological notices (Gratus eleven years, Pilate ten), Josephus' text actually makes sense as it stands, in its current order, in which case Gratus actually governed for four years, Pilate for 17. They both point out that Gratus and Pilate are the only two governors of Judea for whom Josephus gives us such numbers at all (he does not tell us how long Festus governed, for example), and that a motive for adding those two chronological notices can be found in the urge to contradict the spurious Acts of Pilate being circulated under Maximinus..like when Mary writes:
" Josephus does not say Gratus had two terms of office in Judea - but he also did not give the date for the death of Germanicus in 19 c.e.....an omission that contributed to the 26 c.e. dating for Pilate."
I don't know how failing to give the 19 ad date for Germanicus contributed to dating Pilate's appointment to 26 ad, unless, as I guessed, somehow Josephus had linked the two directly.
Feel free to disagree, by all means. It is an hypothesis, not an established theory of the text. Just make sure you know what you are disagreeing with.