On the average run length to false alarm in surveillance problems which possess an invariance structure

Surveillance can be based, in some change-point detection problems, on a sequence of invariant statistics. Gordon and Pollak prove that, under certain conditions, the average run length (ARL) to false alarm of invariance-based Shiryayev-Roberts detection schemes is asymptotically the same as that of the dual classical scheme that is based on the original sequence of observations. In this paper we give alternative conditions under which the two ARLs coincide and demonstrate that these conditions are satisfied in cases where Gordon and Pollak's conditions are difficult to check.