Ordered that the order is affirmed, with costs.
The defendant Metropolitan Transportation Authority of the State of New York satisfied its prima facie burden of establishing its entitlement to judgment as a matter of law by demonstrating that it did not own or operate the subject bus and that it is not vicariously liable for the torts of its subsidiaries such as the MTA Bus Company ( see Public Authorities Law § 1266; Delacruz v Metropolitan Transp. Auth., 45 AD3d 482, 483; Towbin v City of New York, 309 AD2d 505; Emerick v Metropolitan Transp. Auth., 272 AD2d 150; Noonan v Long Is. R.R., 158 AD2d 392, 393; Cusick v Lutheran Med. Ctr., 105 AD2d 681; Dixion v New York City Tr. Auth., 24 Misc 3d 1227[A], 2009 NY Slip Op 51645[U], *4 [2009]). In opposition, the plaintiffs failed to raise a triable issue of fact as to the doctrine of equitable estoppel ( see Delacruz v Metropolitan Transp. Auth., 45 AD3d at 483; Sew Wai Yong v City of New York, 41 AD3d 212, 213; Zaiman v Metropolitan Tr. Auth., 186 AD2d 555, 556-557).
The Supreme Court properly denied the plaintiffs' cross motion for leave to amend the complaint to add the MTA Bus Company as a defendant under the relation-back doctrine ( see Smith v Garo Enters., Inc., 60 AD3d 751, 752; Rinzler v Jafco Assoc., 21 AD3d 360, 362; Lopez v Metropolitan Transp. Auth., 267 AD2d 359; Bartnicki v Centereach Fire Dept., 222 AD2d 637; Zaiman v Metropolitan Tr. Auth., 186 AD2d at 557).