Nonanticipating estimation applied to sequential analysis and changepoint detection

Abstract

Suppose a process yields independent observations whose distributions belong to a family parameterized by θ∈Θ. When the process is in control, the observations are i.i.d. with a known parameter value θ₀. When the process is out of control, the parameter changes. We apply an idea of Robbins and Siegmund [Proc. Sixth Berkeley Symp. Math. Statist. Probab. 4 (1972) 37–41] to construct a class of sequential tests and detection schemes whereby the unknown post-change parameters are estimated. This approach is especially useful in situations where the parametric space is intricate and mixture-type rules are operationally or conceptually difficult to formulate. We exemplify our approach by applying it to the problem of detecting a change in the shape parameter of a Gamma distribution, in both a univariate and a multivariate setting.