Average run length to false alarm for surveillance schemes designed with partially specified pre-change distribution

Observations are taken independently and sequentially. Detection of a change in distribution is studied when the problem has an invariance structure. The prechange distribution is assumed to be a member of a specified family and is assumed known up to a nuisance parameter. We provide a general method of constructing surveillance schemes in the presence of a nuisance parameter and give sufficient conditions for approximating their average run lengths to false alarm. Applications include detecting a change in scale of i.i.d. gamma variates with unknown initial scale, detecting a change in location of i.i.d. normal variates with unknown initial mean, and a non-parametric scheme based on ranks for detecting a change to a stochastically larger distribution.