Minimax optimality of the Shiryayev-Roberts change-point detection rule

The result of Pollak [1985. Optimal detection of a change in distribution. Ann. Statist. 13, 206-227] proving the asymptotic optimality in sequential change-point detection of a suitable Shirayayev-Roberts stopping rule up to terms that vanish in the limit is generalized from the case of two completely specified distributions to that of a composite alternative hypothesis in a multidimensional exponential family. An explicit asymptotic lower bound on the expected Kullback-Leibler information required to detect a change-point is derived and is shown to be attained by a Shirayayev-Roberts stopping rule.