On a generalized disorder problem

Tomasz Bojdecki and Jerzy Hosza

Stochastic Processes and their Applications, 1984, vol. 18, issue 2, 349-359

Abstract: Suppose that independent identically distributed quantities [eta]1, [eta]2, ... appear consecutively. At some random moment [theta] the distribution of the [eta]'s changes to one of the laws [mu]1, ..., [mu]d, and from then on the quantities appearing are distributed according to this new law. Our objective is to find a stopping rule based upon the past values of the [eta]'s only which maximizes the probability that the moment of stopping belongs to a given neighbourhood of [theta]. No restrictions are imposed on the distribution of [theta], and the probability (a priori) that [eta]j is the 'disordered' law may change in time. It is proved that an optimal stopping rule always exists and its form is derived. The maximal probability corresponding to the optimal stopping rule is found as well.

Date: 1984
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