Maximizing the duration of owning a relatively best object

A version of the secretary problem in which the objective is to maximize the time of possession of a relatively best object is treated. The basic no-information model is related to the Presman-Sonin best-choice problem with a random number of objects. A best-choice duration model similar to one suggested by Gaver is also solved. Extensions to models with a discount, to full-information models, and to problems with and without recall are made. Finally, the duration versions of the random arrival models of Cowan, Zabczyk and Bruss in the no-information case and of Sakaguchi and Bojdecki in the full-information case are solved.