On the disorder problem for a negative binomial process. (English) Zbl 1328.60102

Buonaguidi, Bruno; Muliere, Pietro

J. Appl. Probab. 52, No. 1, 167-179 (2015).

Summary: We study the Bayesian disorder problem for a negative binomial process. The aim is to determine a stopping time which is as close as possible to the random and unknown moment at which a sequentially observed negative binomial process changes the value of its characterizing parameter \(p \in (0, 1)\). The solution to this problem is explicitly derived through the reduction of the original optimal stopping problem to an integro-differential free-boundary problem. A careful analysis of the free-boundary equation and of the probabilistic nature of the boundary point allows us to specify when the smooth fit principle holds and when it breaks down in favour of the continuous fit principle.

MSC:

60G40	Stopping times; optimal stopping problems; gambling theory
62C10	Bayesian problems; characterization of Bayes procedures
62L10	Sequential statistical analysis
35R09	Integro-partial differential equations
35R35	Free boundary problems for PDEs

Keywords:

negative binomial process; Bayesian disorder problem; stopping time; optimal stopping problem; integro-differential free-boundary problem; negative binomial process; smooth fit principle; continuous fit principle; regular boundary; sequential detection